Pedro's stunning early goal was soon forgotten as Abdoulaye Doucoure smashed in an equaliser and Roberto Pereyra netted either side of two astonishing Richarlison misses that ought to have put Marco Silva's side in the clear.
And the Blues made the most of the let-off as Michy Batshuayi bagged a brace and Cesar Azpilicueta scored to take Chelsea above their visitors and into the top four.
Coming into the game on the back of bruising defeats to Manchester City and Crystal Palace, and given the volatility of the Stamford Bridge hot seat, some had questioned Conte's long-term prospects, but the Italian says he is not feeling the heat from owner Roman Abramovich.
"I must be honest. I feel a lot of pressure but not this type of pressure if the club decide to sack me," Conte told a news conference after Saturday's win.
"Honestly I don't feel this pressure. I trust in my work.
"I try to keep everything for the club that I'm working for. I'm doing this for Chelsea. I did this last season. I'm doing this with my staff this season.
"Honestly, I will never be worried for this. Never.
"The pressure is I want for my players and to give satisfaction for my fans. I feel this type of pressure.
"Our fans, they show me and my players every game a lot of patience. They show they want to push us in every game.
"I feel this type of pressure, other pressure is not important for me. I don't feel because this is our job.
"When I go to my house I am happy because I know I am putting 150 per cent for my club and my players are doing the same."
The Hornets beat Arsenal 2-1 last weekend, but were denied another high-profile scalp.
Silva refused to lay the blame at Richarlison's feet, despite the Brazilian prodigy's pair of misses from inside the six-yard box.
"He deserved all the credit for what he did until now," Silva said. "It is fantastic.
"I am sure in the next games he will score again. He scored really important goals for us in the past.
"It is more important we created the chances. Richarlison was in the right position to score again."