The Hornets were on course to extend their unbeaten league sequence to eight matches after themselves coming from a goal down to lead 2-1 at the interval.
But Sky Blues striker Best capitalised on two lapses in a defence already without the services of captain Jay DeMerit and on-loan Manchester United youngster Craig Cathcart, through injury and illness respectively, and which was then forced to reshuffle again before conceding their third.
After Will Hoskins had missed a great chance to put the home side in front inside 70 seconds, it was the Sky Blues who struck first after 14 minutes.
The Hornets blocked one attempt but failed to close Sammy Clingan down quickly enough as he broke on to a loose ball to beat keeper Scott Loach with a low 20-yard drive.
The Hornets were back on terms seven minutes later when Stephen Wright could only head a Don Cowie cross on to Tom Cleverley, who fired a first-time left-footed half-volley through the legs of team-mate John Eustace to equalise.
And by half-time the in-form Hornets were in front thanks to a classic counter-attack.
After the ball had pin-balled around their own area, it was cleared to halfway where Hoskins attacked the Coventry defence at pace before slipping in Danny Graham. His attempt was blocked but the ball rebounded perfectly for Hoskins to beat Westwood from 12 yards.
But for the Coventry keeper though, Watford would surely have had the game won by half-time as Graham was denied what looked an odds-on third for the home side.
And the Hornets were made to pay for their failure to wrap the match up midway through the second period.
Although Loach managed to block a header from Aron Gunnarsson, the home side's defending was shambolic as Elliott Ward headed back across the six-yard box for Best to finish with ease from close in.
Best was in the right place again to score what proved to be the winner with five minutes remaining after Clinton Morrison had made a sufficient nuisance of himself to prevent Loach from getting to Clingan's inswinging ball into the area.
But it was Westwood who was to prove Coventry's saviour again late on. After tipping over a Eustace header he then performed heroics to thwart Adrian Mariappa from close range after Isaac Osbourne had blocked another Watford attempt on the line, and then Liam Henderson blasted the follow-up over.