Tim Sherwood's Villa face Liverpool in Sunday's semi-final, having beaten rivals West Brom in the last eight to book a trip to Wembley.
Villa boosted their chances of retaining their Premier League status by drawing with QPR and then beating Tottenham last week.
The Midlands club now have a six-point buffer to the drop zone, but are not out of the mire yet.
Speaking to Perform, Vassell said: "Yeah definitely [they can win the FA Cup]. I've seen Villa on a number of occasions perform well against the big teams and it's the cup so anything can happen.
"They've definitely got an opportunity to win it, they shouldn't be thinking they can't win it. The minute you get in to the finals of any competition anything can happen.
"They'll be doing the best to bring the trophy back to Birmingham but I think first and foremost in the Premier League they need to get well out of trouble.
"I think the cup run and the way they've gone about winning matches in the cup should give them confidence in the Premier League matches, but it could work against them with focus going on the cup. Premier League status is more important."
Vassell, who came through the youth ranks at Villa Park before departing for Manchester City in 2005, has been impressed by the job Sherwood has done since taking over in February.
"I thought he was a good appointment, he's someone bright, young and relishing the challenge," he added.
"Some managers may have waited to see how Villa's fate turned out, but he's come in and given the team a boost. I like the attitude of the players and the way they're fighting for survival."
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