Villa led 2-0 at halftime through goals by Richard Dunne and Darren Bent, prompting near silence from the home fans for long periods.
But a revitalized Arsenal side got back in the game through Robin van Persie’s 54th-minute penalty, equalized through Theo Walcott’s lucky rebound and took the lead in the 61st through a second spot kick from Van Persie.
Having avoided a fourth straight defeat, the Gunners will face the winner of a replay between Sunderland and Middlesbrough next month.
Arsenal won the last of its 10 FA Cups in 2005. It has not won a trophy since.
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