Alexis Sanchez gave Arsenal a controversial fifth-minute lead when his goal was allowed to stand despite the ball striking his arm in the build-up.
Victor Moses was sent off after 68 minutes, receiving his second booking for simulation in the penalty area, but Diego Costa equalised before Aaron Ramsey headed home a 79th-minute winner.
Conte could not secure a double in his first season in charge at Stamford Bridge and singled out Ramsey’s goal and the dismissal of Moses as the turning points.
“I think our start for sure wasn’t good enough,” he said. “Arsenal started the game better than us, honestly I think the goal after three minutes in this way, I think it was not disallowed because there is a clear handball but our start wasn’t good.
“Then we started to play, in the second half we started very well and created the chances to equalise. The second yellow card for Moses was for sure one of the two key moments.
“But to play for 10 men, we were able to equalise and then to concede a second goal was a key moment for the loss.”
The former Juventus coach also refuted claims his players struggled to raise themselves for the final having already secured the Premier League title.
“It is not easy but I think that we won the league and it was right to celebrate and enjoy this win,” he added. “I think that my players put all of themselves into the game, maybe our start wasn’t good and the impact and approach wasn’t good and Arsenal pushed a lot. We paid for this start.”