The Argentine exchanged words with his manager as he was substituted against the Tykes, having spent several minutes asking to be withdrawn from the game due to injury.
Tevez came off in the 78th minute, having asked several times to be withdrawn and limping from the field, but, despite a minor confrontation between the striker and his manager, Mancini was quick to assert that he was "just tired".
The 29-year-old had a hand in all five of the goals, scoring the opener, followed by another two either side of half time, before setting up Kolarov and Silva to cap off a match-winning display.
"We wanted this [performance from Tevez]," he told ESPN. "I'm very sorry for Edin [Dzeko] because in this game it is very important that both strikers score goals. Sometimes this is impossible, like today for Edin. But three goals for Carlos [Tevez] are good for him.
"Was he [Tevez] injured? No, no, no no. Nothing, nothing. I ask him if he had a problem in his muscle or his hamstring, he said no, he was only tired."
The Italian also hailed his side's performance, lauding its concentration against a Tykes side, which the Italian believed "had nothing to lose" in the tie.
"I was very pleased because after the game is easy, but before the game is never easy in football," he continued. "For this reason it is important to have good concentration for these games. The other team have nothing to lose in these games.
"We scored very quick, three in the first half so after, it was finished. When you score three in the first half you need to continue to play serious."