The equaliser came from a knockdown from Oli McBurnie, and then Carvalhal's tactical switch paid dividends again, as fellow substitutes Narsingh and Nathan Dyer combined for the victor.
Only West Brom are below Swansea, and they play Arsenal on Sunday.
It was an incredible result for Carvalhal's first game in charge of the club, and they ended the day in 19th place on 16 points.
Stefano Okaka brought down Heurelho Gomes's goal kick skillfully before weighting a flawless pass inside the left channel to Richarlison.
"At this level, when you don't score, it gives them belief".
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"We didn't play the first 45 minutes as we wish because we had some problems to control".
First, Ayew stayed just onside to tap home a headed assist from McBurnie - on as a 57th-minute replacement for Abraham - before Narsingh sent the visiting fans into delirium after Gomes had pushed the ball back into the danger zone from Dyer's long-range strike.
Watford boss Marco Silva was disappointed and rued Gray's miss as Watford squandered an opportunity for a second successive league victory following a six-game winless run.
Swansea's next game is against Tottenham on Tuesday, 2 January and, despite the euphoric nature of the win at Watford, Carvalhal struck a cautious tone. The last 15 minutes we solved this problem.
The Portuguese only lost his job at Championship Sheffield Wednesday on Christmas Eve, but four days later found himself in the Premier League.