Villa had a stress-free night after El Ghazi struck in the 11th minute and Traore quickly doubled the advantage with a delicious volley.
Watkins, on as a 70th-minute substitute for Traore, scored his second goal in this season's competition within three minutes of his arrival as Dean Smith's side set up a fourth-round home tie against Stoke.
This was a meeting between two teams who had been perfect during the opening weeks of the season.
Championship high-flyers City had recorded four straight wins in league and cup, while Villa had matched that 100 per cent return with victories over Burton, in the Carabao Cup, and Sheffield United in their Premier League opener.
Smith named a completely different line-up to the one that had beaten the Blades on Monday, with John McGinn, Tyrone Mings and Watkins among those benched and no place in the matchday squad at all for Jack Grealish.
Traore started as expected and the former Chelsea striker lined up on the right side of a three-man attack with El Ghazi and Keinan Davis.
Despite the 11 changes, there was nothing disjointed about Villa's start with Max O'Leary producing a fine double stop to deny Davis and Frederic Guilbert.
The pressure was rewarded after 11 minutes when Davis reached the byline and calmly rolled the ball into the path of El Ghazi, who finished with an assured tap-in.
The lead was doubled when Traore superbly cushioned Jacob Ramsey's pass into the top corner with a minimum of fuss.
City were shell-shocked and continued to toil, but Dean Holden's side should have halved the deficit on the half-hour mark.
Niclas Eliasson's cross found Famara Diedhiou unmarked eight yards out, but the Senegal striker planted his header wide of Jed Steer's right-hand post.
It was a rare moment of respite for the hosts and Villa had strong claims for a penalty turned down after 35 minutes.
El Ghazi raced onto Kortney Hause's pass and squared to set up Ramsey six yards out for what would have been the teenager's first senior goal.
But Ramsey fell under contact from Zak Vyner and, much to Villa's disbelief, referee James Linington waved play on.
City showed more endeavour in the second half but struggled to fashion opportunities against a defence marshalled by captain-for-the-night Ahmed Elmohamady and where Hause caught the eye.
The tie was put beyond doubt when Villa sliced through a tiring home midfield and Mahmoud Trezeguet delivered a low cross which Watkins steered home at the far post.