The Roma captain scored a great free kick, a penalty and then missed from the spot before Aleandro Rosi stole it for the visitors after relegated Bari went ahead twice in the first-half.
"I am happy to have got past another record and I hope to continue at these levels," he told Sky Sport Italia after the game.
"It was a thrilling night, but it's a shame about the missed penalty. However, we showed strength and will to react like a great team. The group is united and we want to show we are still alive."
Totti is keeping eyes on Monday night's fixture between Lazio and Juventus as the race for fourth places reopens.
"If we had drawn it would have been over, but now we can pressure them," he concluded.