Roy Keane's side forged ahead through Andros Townsend but the Swans bounced back from their defeat to Portsmouth to pick up victory courtesy of Craig Beattie`s brace and a scrambled strike from Joe Allen.
"I've got a great belief in the group and the players and we've shown we can be competitive," said Rodgers. "If we're still there in February or March, because of the way we play we'll have a very good chance."
Referee Andy D'Urso played a key role in the 88th minute when - with the score poised at 2-1 - he turned away calls for a penalty when Carlos Edwards appeared to be tripped. Swansea then broke away and killed the game off within 30 seconds.
Rodgers added: "If you're the home manager you're disappointed if you don't get it. But having seen a replay, I can see why it wasn't given. There was a slight touch but maybe he (the referee) is thinking their guy can go through, get his shot in and finish.
"There's no doubt there was contact but I don't think it was a turning point in the game. There's no doubt we got a bit of luck in that moment but I don't think anyone could begrudge us the win."
Source: PA
Source: PA