Batshuayi came off the bench to make it 2-2, with Watford having led for much of the second half through Abdoulaye Doucoure and Roberto Pereyra, before Azpilicueta and the Belgium worldwide secured the win in the dying minutes.
It represented a big game in which the Blues could overtake the fourth-placed Hornets, who had won three and lost one of their last Premier League fixtures.
Worse followed for Chelsea when Roberto Pererya put Watford ahead four minutes in the second half.
Tom Cleverley, who scored the victor last week against the Gunners should start in a central role in behind the striker, while captain Troy Deeney is expected to come into this starting XI in place of Andre Gray.
But Batshuayi headed home 20 minutes from time to spare Chelsea's blushes and added his second in the dying moments following an 87th-minute goal from defender Cesar Azpilicueta.
Hazard collected a short corner from Cesc Fabregas and pulled the ball back into the path of Pedro, who whipped in a curling, right-foot shot that clanged into the net off Heurelho Gomes's right-hand post.
In fact it is the first time Eden Hazard, Alvaro Morata and Pedro have started together for Chelsea.
Watford leveled matters on the stroke of halftime as Abdoulaye Doucoure volleyed home after Chelsea failed to clear their lines.
Tiemoue Bakayoko had been a doubt before the game and the Chelsea midfielder, poor throughout, was slow to react when he was dispossessed by Pereyra in the middle of the pitch.
Second half substitute, Batshuayi equalisd with a superb header!
Batshuayi made it four in added time after a terrible error by the Watford defence to take the Blues above their visitors to 4th place on the league.