Despite dominating the match from start to finish, Arsenal needed a goal four minutes from time - a firecracker from Wilshere - to earn a 1-0 win and book a trip to Brighton in the fourth round.
The Gunners blew a host of chances, but Wenger said he was always confident his side would, in the end, find away through.
"I always had faith" he said. "I always felt maybe one (chance) will go in. It was unbelievable that we didn't score (earlier). At least we scored before the end."
Wenger says Wilshere's return to full fitness after his long lay-off with injury is a huge boost to the team.
And against Swansea the 21-year-old excelled in a more attacking role than usual.
"What is pleasing is that he's ahead of schedule. Jack is completely back to his best and I would never have believed that would have happened as soon as January," he said.
"He can play everywhere - he can defend and attack. He has got back his burst with the ball so I used him closer to the strikers and he was always dangerous.
"A complete midfielder can defend and attack and Jack can also dribble and give a final ball so the closer players like him are to the goal the better it is for them.
"He has quality and enthusiasm and a love for the game, which is the most important thing for me."