Inter keeper Julio Cesar failed to hold a long-range shot from Arjen Robben and Gomez pounced on the rebound to score in the final minute of normal time, leaving last year's champions with a huge task to avoid an early exit.
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Bayern, 2-0 losers in Madrid last season to the team then coached by Jose Mourinho, looked more dangerous throughout the last-16 first-leg tie, although the hosts also created openings.
"Inter surprised me," Bayern coach Louis van Gaal told reporters. "They played to win and that is very beautiful. The whole world has seen a wonderful game and I think both sets of fans could be happy with this game.
"That is what football is about and having enjoyment in football. In the final last year, they attacked less than they today and got a better result. But that's football."
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Inter had an early chance when defender Andrea Ranocchia side-footed wide from a free kick but Bayern then looked more threatening.
Luiz Gustavo was twice close while Franck Ribery, always dangerous with the ball at his feet, headed against the crossbar from Robben's curling cross.
Samuel Eto'o was Inter's most threatening player, having a penalty appeal turned down, creating a good chance for Esteban Cambiasso and then producing a brilliant reflex save from Thomas Kraft before halftime.
Fullback Maicon had Inter's best first half chance in stoppage time when was given a clear shot on goal but fired wide.
Robben continually tormented Inter and hit the post early in the second half, then struck another effort over the crossbar after cutting inside with one of his trademark runs
In between, Inter missed a sitter when Eto'o had another shot saved by Kraft and Cambiasso, unmarked, fired over from the rebound.
Inter finished with a flourish as Thiago Motta was foiled by Kraft and an Eto'o shot went agonisingly wide.
"Bayern had more ball possession but our chances were better than Bayern's," Inter coach Leonardo said.
