The hosts took a two-goal lead through Jordon Mutch and former Sunderland striker Fraizer Campbell.
But Poyet's team responded superbly and hit back with two late goals from substitutes Steven Fletcher and Jack Colback.
Speaking to Sky Sports after the game, Poyet said his side were quite simply too slow out of the blocks.
Poyet said: "It was an incredible day, but we started poorly. It looked after they scored their second that the game was dead.
"We fought hard though and put Cardiff on the back foot and in the end I thought we deserved the equaliser. Mentally in football it's vital you approach the game in the right way and I don't think our warm-up was a good one.
"It took us 30 minutes to get going and understand that you need to battle and we did that in the end."
Fletcher was introduced at the break after Fabio Borini was forced off with an illness and Poyet thought he made a big difference.
Poyet added: "Fletcher is a player who is difficult to find. He is very important to us but it is impossible to play him all the time. He works very hard.
"We try to get the best out of him and realise he is probably more dangerous up front in a two.
"We are five games unbeaten now and this is very important because it means we are difficult to beat.
"You must remember that some of our defenders today haven't played together for three months."