The Premier League leaders were held to a 1-1 draw at The Hawthorns in their previous top-flight fixture as Victor Anichebe popped up with an 87th-minute equaliser for the struggling hosts.
Chelsea looked set to drop points again on Saturday as Roberto Martinez's Everton held firm at Stamford Bridge while creating opportunities of their own.
Yet Mourinho's men made a vital breakthrough in injury-time, fit-again captain John Terry being credited with the goal after challenging Everton goalkeeper Tim Howard from a Frank Lampard free-kick.
Reflecting on his side's dramatic triumph, Mourinho said: "The feeling when you lose points in the last minutes, like we lost two points against West Brom, is a bad feeling. When you win it's the opposite. It's basically a compensation of the points we lost against West Brom.
"'For the last 15 minutes we were strong; we created; we pressed; we dominated. When you score (a late goal) you can speak about luck ... but the reality is the boys chased that.
"A point for them (Everton) is good. A point for us is not good, and in the end maybe we deserved it."
Chelsea's triumph ensured that they will finish the weekend as they started it, at the top of the Premier League table.
With Arsenal and Manchester City both in action after his side this weekend, Mourinho was delighted to have tasted success.
"It was a very difficult match. In periods one team was dominant and the other team had to play well very defensively to cope with that," the Portuguese added.
"If we get a draw, I think in a couple of hours we are third in the table (Chelsea could actually only slip to second, although City have a game in hand). This is the reality in this competition.
"Every victory keeps us there (at the top of the table). Every defeat or draw, we lose a position.
"Of course it's a good feeling to win in the end. We have some limitations, that's obvious, but the team was fantastic."
One negative for Chelsea was an injury that forced Oscar to be withdrawn at half-time.
"He said he could cope with the pain, there was no risk to make it worse, but he couldn't," Mourinho explained.
