Exhibit 10.1

AMENDMENT NO. 2

DUNE ENERGY, INC.

2012 STOCK INCENTIVE PLAN

NON-QUALIFIED STOCK OPTION AWARD AGREEMENT

Dated: January 15, 2013

WHEREAS, Dune Energy, Inc. (the “Company”) previously granted, pursuant to the “Dune Energy, Inc. 2012 Stock Incentive Plan Non-Qualified Stock Option Award Agreement” (the “Agreement”) a non-qualified stock option to Stephen Kovacs (“Participant”).

WHEREAS, Participant resigned from his service to the Company as a member of the Company’s Board of Directors effective January 8, 2013 (the “Resignation Date”).

WHEREAS, the Company and Participant desire to amend the Agreement to provide (i) that the Option shall vest in its entirety as of the Resignation Date, (ii) that Participant shall have two years from the Resignation Date to exercise the Option, and (iii) for a mutual release of claims between the Company and Participant.

NOW, THEREFORE, effective as of the Resignation Date, the Agreement is amended as follows:

1. Section 5(b) and Section 5(c) shall be deleted and replaced in their entirety as follows:

(b) Termination for any other reason. In the event of the Participant’s Termination, resignation, or removal for any reason (except for death, Disability or Cause), the vested portion of this Option shall remain exercisable until two years from the date of Termination.

(c) Treatment of Unvested Option Shares upon Termination. Any portion of this option that is not vested as of the date of the Participant’s Termination for any reason shall immediately become vested upon the date of such Termination.

2. A new Section 21 shall be added as follows:

21. Mutual Release of Claims.

(a) Release by Participant. Participant, for Participant and for Participant’s affiliates, successors, heirs, subrogees, assigns, principals, agents, partners, employees, associates, attorneys, and representatives, voluntarily, unconditionally, irrevocably and absolutely releases and discharges Company, and any parent and subsidiary corporations, divisions and affiliated corporations, partnerships or other affiliated entities of Company, past and present, as well as Company’s employees, officers, directors, agents, successors and assigns, from all claims related in any way to the transactions or occurrences between them to date, to the fullest extent permitted by law, including, but not limited to, Participant’s service to the Company, the termination of Participant’s service, and all other losses, liabilities, claims, charges, demands and causes of action, known or unknown, suspected or unsuspected, arising directly or indirectly out of or in any way connected with Participant’s service to the Company. This release is intended to have the broadest possible application and includes, but is not limited to, any tort, contract, common law, constitutional or other statutory claims, including, but not limited to alleged

violations of the Texas Labor Code (including but not limited to the Texas Civil Rights Act, the Texas Payday Act, and the Texas Minimum Wage Law), Title VII of the Civil Rights Act of 1964, the Americans with Disabilities Act, the Family and Medical Leave Act, the Age Discrimination in Employment Act, and all claims for attorneys’ fees, costs and expenses. Participant expressly waives Participant’s right to recovery of any type, including damages or reinstatement, in any administrative or court action, whether state or federal, and whether brought by Participant or on Participant’s behalf, related in any way to the matters released herein. However, this general release is not intended to bar any claims that, by statute or other applicable law, may not be waived. Notwithstanding the foregoing, nothing within this paragraph releases Company from the promises it has made herein in this Agreement. Participant acknowledges that he may discover facts or law different from, or in addition to, the facts or law that he knows or believes to be true with respect to the claims released in this Agreement and agrees, nonetheless, that this Agreement and the release contained in it shall be and remain effective in all respects notwithstanding such different or additional facts or the discovery of them. Participant declares and represents that he intends this Agreement to be complete and not subject to any claim of mistake, and that the release herein expresses a full and complete release and the parties intend the release herein to be final and complete. Notwithstanding anything in this Agreement to the contrary, nothing in this Agreement shall be deemed to constitute a release by Participant of claims against the Company relating to the Company’s obligations to indemnify Participant in connection with Participant’s service to the Company as a director.

(b) Release by Company. The Company, for the Company and for the Company’s affiliates, successors, heirs, subrogees, assigns, principals, agents, partners, employees, associates, attorneys, and representatives, voluntarily, unconditionally, irrevocably and absolutely releases and discharges Participant, as well as Participant’s affiliates, successors, heirs, subrogees, assigns, principals, agents, partners, employees, associates, attorneys, and representatives, from all claims related in any way to the transactions or occurrences between them to date, to the fullest extent permitted by law, including, but not limited to, Participant’s service to the Company, the termination of Participant’s service, and all other losses, liabilities, claims, charges, demands and causes of action, known or unknown, suspected or unsuspected, arising directly or indirectly out of or in any way connected with Participant’s service to the Company. This release is intended to have the broadest possible application and includes, but is not limited to, any tort, contract, common law, constitutional or other statutory claims, and all claims for attorneys’ fees, costs and expenses. The Company expressly waives the Company’s right to recovery of any type, including damages or reinstatement, in any administrative or court action, whether state or federal, and whether brought by the Company or on the Company’s behalf, related in any way to the matters released herein. However, this general release is not intended to bar any claims that, by statute or other applicable law, may not be waived. Notwithstanding the foregoing, nothing within this paragraph releases Participant from the promises it has made herein in this Agreement. The Company acknowledges that it may discover facts or law different from, or in addition to, the facts or law that it knows or believes to be true with respect to the claims released in this Agreement and agrees, nonetheless, that this Agreement and the release contained in it shall be and remain effective in all respects notwithstanding such different or additional facts or the discovery of them. The Company declares and represents that it intends this Agreement to be complete and not subject to any claim of mistake, and that the release herein expresses a full and complete release and the parties intend the release herein to be final and complete.

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IN WITNESS WHEREOF, the Company and Participant have executed this Amendment No. 2 as of the date first written above.

IN WITNESS WHEREOF, the Company and Participant have executed this Amendment No. 2 as of the date first written above.