City, trailing 1-0 from the first leg, dug themselves a much deeper hole as Mathias Fernandez and Ricky van Wolfswinkel put Sporting 2-0 up in the first half.
But City got themselves back into it in a thrilling final half-hour as Sergio Aguero scored either side of a Mario Balotelli penalty.
In the last of five minutes of injury-time, Hart - up for a corner - got on the end of a cross from Nigel de Jong and sent a strong header goalwards, only to see his opposite number touch it narrowly wide.
"I can't believe it, I thought it was in," Hart told ESPN.
"It didn't need to get to that, but football is football. The fans were fantastic tonight [Thursday] and we didn't give them what they deserved in the first half.
"Hopefully we can take some positives out of the second half and move on for the rest of the season."
City now have extra time to recover with their next league game pushed back to Wednesday due to opponents Chelsea playing in the FA Cup on Sunday.
"We need to play well in every game," Hart added. "There are not many left now. We've cost ourselves with the first half but if we keep rolling now we'll be all right.
"We're one point behind the leaders of the Premier League, we're a good team and we've got nothing to worry about. We'll go in (against Chelsea) as favourites and we should be favourites."
Kolo Toure, captain in the absence of Vincent Kompany, said City had no time to rue their exit as they look to catch Manchester United in the league.
"It was important to win this game and go through, but now we need to keep our heads up," he said.
"We have a really important game coming up on Wednesday against Chelsea and we have to give it all.
"For Man City now every trophy is very important so when we go out like that, it is always very difficult, but now we just need to think about the future. The past is the past."