On a dramatic night at the Boleyn Ground, Enner Valencia gave his side the lead before Kevin Mirallas equalised with just eight minutes of normal time remaining.
Romelu Lukaku thought he???d won it in extra time for the 10-men of Everton before Carlton Cole equalised for the hosts, who went on to win the shootout thanks to goalkeeper Adrian???s winning penalty.
“It was unbelievable,” Allardyce said.
“The highs and lows in one particular game. It has been a long time since I experienced such highs and lows in one match. Such excitement. It was end to end from the first whistle – nothing like a Premier League game.
“Both teams went out and played and tried to win it. In the end we got what we deserved, there is no doubt about that.
“Unfortunately we missed too many chances, their goalkeeper made some fantastic saves. We had 23, 24 attempts. We missed and fluffed a few and had to rely on penalties in the end to win it.???
West Ham had not beaten Everton in 15 attempts before tonight and Allardyce thought he was going to be on the wrong end of another result against the Merseyside club when Stewart Downing???s spot kick was saved during the shootout.
But Adrian was the hero of the hour for West Ham, saving Steven Naismith???s penalty and stepping up to score the winning goal after Joel Robles had missed for Everton.
And Allardyce was full of praise for the Spanish keeper after helping book West Ham???s place in the fourth round against Bristol City.
“Everton have been a bit of a bogey team for us and when Stewart Downing missed it I thought, ‘here we go again’,??? he added.
“But Adrian saved one brilliantly and took the last one, he was calm. He was determined and it was a victory we deserved over the two legs.”