The champions took the lead at Old Trafford on Sunday through Willian but goals from Romelu Lukaku and Jesse Lingard secured the victory for the home side.
There was a controversial moment four minutes from the end of normal time, when Alvaro Morata steered Cesc Fabregas' pass into the net only to be flagged offside.
Replays showed it was an extremely close call, with Chris Smalling almost exactly level with the Chelsea striker, and Conte believes the use of a video assistant referee, which has been trialled in cup games in England this season, would have made a difference.
"I think the draw was the fair result for this game. But we are talking about a defeat and that's all," he told Sky Sports.
"I must be honest, I didn't see the situation, but if you're telling me Alvaro Morata wasn't offside, I think VAR is very important, especially in England. There are a lot of moments in the game where there are things to super-check."
The defeat left Chelsea in fifth place in the table, with Tottenham having moved into the top four with a 1-0 win over Crystal Palace, and Conte admits his side are now facing a tough task in the race for the Champions League places.
"We had to be more clinical," he said. "Instead, we kept the game open, and the opponent made us pay. Manchester United are a great team, and we must be disappointed, because I repeat today, we could have done better.
"We know there is a battle [for the top four] and it will be difficult to find a place in the Champions League."
13 - Jesse Lingard has scored 13 goals in all competitions for Manchester United this season; only Romelu Lukaku (22) has netted more for the club. Handy. pic.twitter.com/uga5YvCJpT— OptaJoe (@OptaJoe) February 25, 2018
Midfielder Danny Drinkwater felt Chelsea deserved at least a draw against his old club, telling BBC Sport: "We definitely deserved more; first half, we were dominant. The second half was more even, but we didn't deserve to lose that match, no way.
"Obviously there are bits we can improve on. First of all, we can improve on not conceding, and scoring more goals, but football can sometimes go against you and today's one of those days.
"I didn't see Spurs' game, I was concentrated on ours, but it's down to us now. We have to get our results in order and hopefully the other results will look after themselves."