Substitute Ben Watson snatched victory for Wigan from a 90th-minute corner after Pablo Zabaleta had been sent off nine minutes earlier.
Mancini, chasing his second FA Cup in three seasons, said: "They scored in the last minute so for us it was difficult.
"But we didn't play very well - I don't know why.
"We had a chance because we had two or three chances and created more. But in the end, we didn't play like we do usually."
The Blues travelling army was stunned into silence as Wigan, favourites to be relegated, upset the Wembley odds.
Mancini added: "I am very sorry for the fans, because there were 40,000 of them here."
Vincent Kompany gallantly applauded the Wigan players as they received the Cup.
The City skipper said: "Wigan deserved to win, but we'll be back."