Jack Collison struck both goals in the first-leg 2-0 win.
Allardyce was full of plaudits for his men after their performance against Cardiff.
"We were excellent, we asked the lads to control their emotions on a big night like this, not to cause a problem for ourselves and I thought they handled the pressure excellently and played very well indeed," he told Sky Sports.
"Tactically we did everything right and nullified Cardiff to minimal chances on their own ground and getting forward was outstanding and chances were created, scored and a lot missed.
"It has been a great night for us and it is only 90 minutes of a two-legged affair and anything could still happen, the onus is on us to see it through and there is no pressure on Cardiff and mentally we have to be right for it."