The hosts were well below their best in the first half against the Toffees, but they responded after the break following Sherwood's team-talk.
Emmanuel Adebayor's goal proved to be the difference when he fired the ball past Tim Howard in the 65th minute.
The win has now seen Spurs move above Everton into fifth spot, while they have closed the gap on Liverpool to just three points.
Speaking after the match, Sherwood was quick to praise his side's performance and response after struggling in the first-half.
He told Sky Sports: "Everton never had a shot in the second half. Our game plan was to put pressure on high up the field - it didn't change from the first half, we just didn't apply it well enough.
"We're delighted with the result. Yeah, the performance can be better but it's about character - grinding it out when you're not playing well - and a lot of the games are going to be like that. It was chalk and cheese between the first and second halves, but in the end we've come out on top.
"It was huge, it was a cup final. I don't want to rubbish our performance - but sometimes people win cups when they haven't played very well."