After the hosts had applied virtually all the pressure without breaking the deadlock, it was the Swans who went 1-0 up in the 43rd minute through Luke Moore's deft chip.
Darren Pratley, who left the Welsh outfit for Wanderers in the summer, squared things by heading home an equaliser in first-half stoppage time before Chris Eagles netted the winner 11 minutes after the break, reacting when goalkeeper Gerhard Tremmel could not hold on to Martin Petrov's strike.
Swansea substitute Danny Graham almost secured a replay in the final few minutes with a header that came back off the post and Coyle thought his team should already have put the matter beyond any doubt by that point.
"It was more than a win - I thought we were excellent," Coyle said.
"We started the game really well and felt really hard done by to lose the opening goal, because we had various chances to be in front.
"Swansea have some very good players and that proved to be the case.
"The reaction was great and we went in at half-time level, but we should have been in front.
"Then in the second half we scored a great goal and should have added to it if the truth be told.
"When there is only the margin of a goal between teams, anything can happen.
"Swansea had a chance late on when the lad hit the post, but with all due respect, the game should have been dead and buried long before that.
"We played well and created a lot of chances, but should have scored more goals than we did.
"Aside from all the positives, one thing we would have loved is to be more clinical."
While Coyle only made three adjustments to his first XI for the match, Swansea manager Brendan Rodgers completely reshuffled his pack with 10 changes - and insisted his selection was not disrespectful.
"It was about using the squad," Rodgers said.
"I really wanted us to go through and I felt I had picked the team to do that.
"It certainly wasn't a case of disrespecting the tournament because the FA Cup is the best domestic cup competition in the world."
Bolton forward Kevin Davies, who has lost his regular place in the starting line-up this season and has been linked with a move, received a warm reception from the Wanderers supporters when he came on as a substitute in the second half.
Sunderland have enquired about the 34-year-old, but Coyle remains adamant that Davies has a future at the Reebok Stadium beyond Tuesday's transfer deadline.
Asked if Saturday's cameo would be the last Bolton fans would see of Davies in a Trotters shirt, Coyle said: "I would be very surprised if that was the case.
"There is all sorts of speculation and I think when good players are out of the team, that is always going to happen.
"I would hope Kevin Davies would get a wonderful reception because he is an outstanding player.
"He is a terrific captain and he has a role to play at this football club. He knows that - I have spoken to him extensively.
"I can understand why there would be interest. [Sunderland manager] Martin O'Neill did ring me to make an inquiry, but I told him that Kevin was a part of my plans and that I don't want him going anywhere.
"You can't stop clubs enquiring and that is exactly what it is - there was no offer or anything else."
Regarding potential incoming personnel, Coyle confirmed his interest in Crystal Palace youngster Wilfried Zaha and expressed his hope that he would be able to add to the signing of Tim Ream, the defender recruited from the New York Red Bulls last week, before the window closes.
"There are players out there - and Zaha is certainly one of them - who I think would enhance our club for years to come," Coyle said.
"There are one or two I would love to bring in on a permanent basis or in a loan capacity before the 11pm deadline on Tuesday.
"I won't stop targeting good young players to bring into this football club who I think I can develop and make better."
