England Under-21 striker Leroy Lita, making his first start of the season, nudged home Reading's winner midway through extra-time.
But the major talking point of their 1-0 win was defender Sam Sodje's 58th-minute dismissal for a second yellow card.
Sodje, who is believed to be a strong transfer target for Sheffield Wednesday, departed the action following a foul, leaving Coppell simmering.
He said: "I try to bite my tongue sometimes because I don't like complaining about referees, but I did feel there was an element of unfairness in some of the decisions.
"I didn't feel the consistency for both sides was right. The referee reacted to the crowd's reaction for the second tackle.
"It is a hostile atmosphere here (Liberty Stadium), and you need your officials to be strong.
"I just felt we were on the wrong end of quite a few decisions.
"Sodje's hardly made a tackle, and then as a centre-half, he is strong and wins the ball.
"To be punished with the foul and then to be given a second yellow card - I thought it was outrageous."
League One outfit Swansea had by far the better chances of an entertaining contest, culminating in Paul Anderson hitting the bar during extra-time.
Lita though, showed the Swans exactly how it should be done, applying a delicate finish to fellow striker Dave Kitson's cross and sending Reading into round three.
Coppell added: "It is nice to have Lita back, but for both him and Kitson to play the full 120 minutes, they did terrifically well to come through it.
"In everything we did, I thought we showed our desire to be in the next round.
"I know people criticise what I do in cup competitions with changing the players around, but we always want to go through.
"We were playing against a very good side. I want to compliment Swansea.
"I had watched a couple of their games, and they play effective football which is pleasing on the eye. I knew it was going to be a difficult test."
Swansea could only reflect on opportunities that went begging, with Anderson apologising to City fans for the succession of missed chances.
But he added: "This is a very good squad, strong enough to push on and hopefully gain promotion."
Sporting Life