Striker Leon Clarke was the toast of Coventry after firing a stoppage-time winner to send them through to a third-round tie at Barnsley.
Clarke's 35th-minute goal look destined to win the League One side's replay against Hartlepool but defender Jack Baldwin drew the visitors level with three minutes to go.
Just as both sides were preparing for extra-time, Clarke was the first to react when Carl Baker's shot was saved by Scott Flinders and he coolly slotted past the impressive goalkeeper.
Cooper said: "I said to the referee that I hoped he was positive Leon Clarke was onside for the first and that it was a free-kick leading to the second. He said he was and I can only accept it.
"We tried every way to get back into the game. When we haven't dealt with the free-kick and they were given the opportunity to put the ball into our box without having to earn it, that is galling from a defensive point of view.
"Scott Flinders keeps coming up with the goods but we haven't needed him that much this season and when we have he produces what we need. That's why we keep saying that we have got a good goalkeeper on our hands.
"We have to move on and we have got Burton Albion in the league on Saturday."