The win maintains Chelsea's 100% record this season after Costa also scored a late victor over West Ham last Monday night.
Etienne Capoue broke the deadlock in the 55th minute when Watford's French midfielder lashed his shot into the top corner from Adlene Guedioura's cross.
Conte is enjoying a flawless start to life at Stamford Bridge and praised the togetherness of his squad after turning to his bench for match-winning inspiration.
He said: "We all know Diego and Diego is a passionate footballer".
Watford have now won only one home game in eight and will have to be careful how they integrate £40m worth of new recruits, with games against West Ham, Arsenal and Manchester United looming.
It was harsh on the Hornets who had looked as though they would pinch a deserved win with a hard-working display. When new players come on, there is a great impact.
"We really want to show the fans we can compete in matches like these, and win them as well".
"When you don't start the game (well), usually you're not happy". The manager must choose the starting XI and then try to find the right solution during the game. In these two games, we saw this.
"I feel good because I saw the right commitment, the right spirit from my team", Conte told the BBC.
"I don't think Real were upset when they broke the record with Gareth Bale or Cristiano". We need the right message for the confidence, for the work from the training sessions. Evans is only 28 years of age and still has a couple of good seasons in him.
"It won't be easy".
"I hope he continues with his attitude, but sometimes when everyone is talking about one player, he is at the centre of attention".
Mazzarri will be hoping for better luck than he has endured in their last six meetings, however, which ended in three draws and three wins for Conte's old club, Juventus. However, the Italian said that the Blues will not be held ransom to exorbitant player transfer fees.
Chelsea sought an immediate response but much of their forward play had lacked conviction and that trend continued with Matic dragging a shot wide from the edge of the 18-yard box. We played very well.