Conte's future at the west London club is still uncertain despite the battling 1-0 victory against Jose Mourinho's United side which started the final as favorite but fluffed its lines.
The crucial moment of the final came in a tepid first half as Chelsea's Eden Hazard made a brilliant surge into the United penalty area and was crudely brought down by England defender Phil Jones.
Jones was perhaps lucky to escape with just a yellow card, but Hazard got up to slot the penalty kick past David de Gea in the 22nd minute for the only goal of the game.
United, which left injured top scorer Romelu Lukaku on the bench, pushed hard for an equalizer by dominating the second half, but could not find a way through.
Thibaut Courtois, recalled by Conte to play in goal after Willy Caballero had played the other games in the cup run, kept United at bay with a string of fine saves, the best from an attempted chip from Marcus Rashford as he rushed through on goal.
Alexis Sanchez had the ball in the net, but it was ruled out for offside under the VAR system, while United's record signing Paul Pogba had the best chance to level eight minutes from time, but headed wide from a corner.
Mourinho, who had never lost an English domestic final, was in typically defiant mood after the defeat.
"We were the second best team in the Premier League and we were the best team in the FA Cup and didn't deserve to officially be second best," he told BBC Sport.
Conte, who was adding the FA Cup to the Premier League title Chelsea won in his first season in charge, was equally defiant despite reports he will be leaving after his side finished a disappointing fifth in the Premier League in this campaign.
"You can tell what you want, but I'm a serial winner," he said.
"I think after two years the club knows me well. I they want to continue to work with me they know me, I can't change," he added.
It was the eighth FA Cup trophy for Chelsea, United had been bidding for a record-equaling 13th, and its first since 2012.
Prince William, who is president of the FA, normally hands out the cup at the final, but with his brother Harry getting married to Meghan Markle earlier Saturday, could not attend.
Instead, the honor was given to the Jackie Wilkins, the wife of former Chelsea and Manchester United great Ray Wilkins, who passed away earlier this year.
While the eyes of the world may have been on events at Windsor about 20 miles away from Wembley, it provided a fitting finale to a match that while not reaching the heights has handed Conte and Chelsea some consolation for an otherwise disappointing season.