Everton assistant boss Steve Round was pleased with how they battled through against West Bromwich Albion.
The Toffees looked like heading out after Chris Brunt's penalty had put West Brom ahead.
But with two minutes remaining Marouane Fellaini volleyed home from the edge of the box to take the tie into extra time.
And just before the end of the first additional period captain Phil Neville produced a brilliant 25-yard strike to clinch victory.
"At Everton you go right to the death, that is one of our strengths with the spirit we have got," said Round.
"They never give up and they keep going and you can go back over countless games where we have come back to get a result and even win.
"Last night was no different. We went right to the end and got our rewards."
The assistant manager added: "We found it difficult to get the breakthrough and we needed to score early.
"We had a lot of possession in the first half but couldn't turn that into chances. You are always susceptible to the sucker punch and we got caught in the break.
"Luckily right at the death Fellaini pops up with a great goal and into extra time Phil Neville produces a wonder-strike to win it.
"We are delighted we are in the hat for the next round."