Graham's haul would include two League Cup successes, the second as part of a domestic cup double alongside the FA Cup in 1993 - Arsenal becoming the first side in English football history to achieve the feat.
But Bravo did well to push Bjorn Engels' header onto the post in the 88th minute and ensure the 60th League Cup final was a victorious one for City.
City are the first side to win the competition for three seasons in a row since Liverpool claimed four straight titles from 1981 to 1984.
Victory for City, a few days after their superb Champions League last-16, first-leg fightback to beat Real Madrid 2-1 at the Bernabeu, came against a backdrop of an appeal against their two-year European ban from Uefa for breaching Financial Fair Play rules.
Manchester City top scorer Sergio Aguero is fit despite being forced off in the Carabao Cup final with a cut. Rodri's cross was headed into Aguero's path by Phil Foden and the Argentine struck into the net.
Despite his success in the competition, Guardiola is even among those calling for its head.
A two-goal lead usually spells the end for City's opponents, but a weird slip from Stones gave Dean Smith's Villa side an unlikely route back into the match.
The Spaniard powered a downward header into the net from Ilkay Gundogan's corner, which left a bitter taste for Villa who felt the set piece should not have been awarded.
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Mbwana Samatta's diving header got Villa back in the contest ahead of the break but City dominated the second half before surviving an almighty late scare.
'I always ask his opinion not only for this game because he obviously knows the club very well, but I think this is the way we work.
We were taking on one of the best teams in the world. We knew we would have to be patient without the ball, they went 2-0 up and it is a mountain to climb against a team of that quality but we almost climbed that mountain'.
"Our concentration now goes onto the Premier League", Smith said.
PHIL FODEN, a surprise inclusion in the Manchester City team, revealed he found out on Saturday he would be involved and admitted he needed to impress when given the opportunity. "At 19 years old you have a lot to improve but with this behaviour he can reach whatever he wants".
'I couldn't wait to play.
Southgate wants players who feature regularly for their clubs and some senior City players are known to be bemused that Foden has not started games more regularly this term given his contribution when he has played and his performances in training. When I get into the team, I have to make the most of it'.