The Brazilian's stunning set-piece set up a mouthwatering fifth-round tie at Manchester City for the Blues, who have won the trophy in four of the last seven seasons. City beat them in last season's semi-final.
Mourinho, whose most recent trophy for Chelsea was the 2007 FA Cup, believes his team deserved to win more handsomely than 1-0 after Oscar and Andre Schurrle also struck the goal frame.
"The free-kick obviously made us happy," Mourinho said.
"It's always good when you see a player that, after the normal training session, dedicates some minutes every day to train a specific thing. To score a beautiful goal was good.
"It's a bit of a contradiction because we played so well and you win with a free-kick. When you play so well you win by scoring some amazing goals in free play.
"The 1-0 doesn't reflect the quality of our game and the dominance of our game.
"We hit the post twice. (Stoke goalkeeper Asmir) Begovic again showed how good he is. The boys played really well.
"We controlled the game since minute one and I'm happy not just with the victory, but also the performance."
It was a win which allowed Mourinho to enjoy his birthday dinner with his wife and two children.
He said: "One thing now is to have a dinner where I don't have to pretend that I'm happy; it's better to have a dinner where I'm happy."
Oscar is Mourinho's chosen playmaker, something which saw Juan Mata marginalised and move to Manchester United.
Mourinho sanctioned the deal with a heavy heart as Mata, Chelsea's player of the year in each of the last two seasons, was unable to satisfy his demands to track back in defence.
The Portuguese, who read Mata's departing open letter to Chelsea fans, said: "I feel sorry that I didn't make him happy, but football is football and I build a team around Oscar in that position and on the sides the other people is doing very, very well.
"Juan is not comfortable on the sides. He tried very hard and he did very well in some matches playing on the sides, but it's not his natural habitat.
"So when a club like Man United comes and for sure they told him he's going to play in his favourite position and he wants to go, he goes."
Eden Hazard is another whose performances have impressed Mourinho.
The Belgian again performed well despite close attention from Stoke.
"Some other clubs, not just in England, especially not in England, they cry a lot with the (lack of) protection of their big players," Mourinho said.
"I don't want to cry, because it's not my nature, it's not Chelsea's nature. But, because Eden is small but a very strong boy and resists a lot and doesn't like to dive and doesn't like to be on the floor, opponents are coming very, very strong on him.
"(Referee) Mr (Chris) Foy was just a little bit late to give the yellow card to (Geoff) Cameron. It was on the third tackle.
"But he gave the yellow card and after the yellow card the situation was different. After that Cameron played a normal game."
The reassuring presence of the re-signed Nemanja Matic in midfield allowed the attacking players to flourish.
"The team flies when somebody makes it so simple," Mourinho said.
"He was very comfortable on the pitch. He stole lots of balls and his pass was always quality."