The England international cut towards the center of the area and curled a shot into the top corner for Arsenal's first goal in six games against Chelsea.
"It was nice to see that one go in," Oxlade-Chamberlain said.
"To come to Wembley against a great team like Chelsea after the season they had last year, it's a tough challenge. But it's those challenges we need to overcome this season."
Chelsea midfielder Ramires had a great chance before half-time but headed over goal from close range.
Eden Hazard then shot over after being sent in by Cesc Fabregas in the second half and Oscar's free-kick was punched away by a lunging Cech.
Arsenal's Olivier Giroud then shot over goal before Cech caught Kurt Zouma's floating header as the champions continued to press without much luck
Chelsea keeper Thibaut Courtois kept his team in the match by coming well off his line to deny Santi Cazolra and Kieran Gibbs after both were put through alone but the Blues weren't able to get the equalizer.
Chelsea open their title defence next Saturday against Swansea, while Arsenal host West Ham the following day.