It was a hard-fought Lower Saxony derby as the third-placed team in the second division took on the third-bottom Bundesliga club, with the slumping Wolves having been league runners-up and German Cup winners only two seasons ago.
Gomez sent Braunschweig goalkeeper Jasmin Fejzic the wrong way with his 35th minute spot kick, although the striker, who had scored 16 league goals in the regular season, controlled the ball with his hand prior to Wolfsburg earning the penalty.
The hosts squandered two more big chances to boost their lead early in the second half through Gomez and Daniel Didavi.
The Wolves faced little resistance after the break from Braunschweig, who had to wait until stoppage time for their first shot on goal in the second period.
The teams will meet again on Monday for the return leg in Braunschweig.
(Reporting by Karolos Grohmann; editing by Ken Ferris)
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