Mourinho named Costa on the substitutes' bench and stated ahead of Saturday's game at Stamford Bridge that it would be a "gamble" to play the Spain international.
The Chelsea boss brought Costa on at half-time after Charlie Adam had cancelled out Eden Hazard's penalty with a sensational strike from his own half just before the break.
Costa lasted only 11 minutes before he pulled up again and it was left to Loic Remy to score the winner for a second game in a row, sending Chelsea seven points clear at the top of the table.
Mourinho admitted it was a risk to bring Costa on, but has no regrets and laid no blame at the door of his medical department.
He told Sky Sports: "If my medical department decides to take four weeks to recover Diego this would not happen, but in football we need risks. I risk sometimes, as a manager in a result I don't like.
"We played more offensive and if the opponent scores I don't kill myself for that.
"But in all the scans they did two days ago the muscle was clear, the image was clear and the player trained twice 100 per cent, when he did much more than he did today.
"We put him in dozens of actions and the result was correct. My medical department is fantastic so there is no problem. Next week plays Remy and (Didier) Drogba, we wait for Costa again. For sure with that injury it will be a couple of weeks minimum."
Mourinho was pleased with the persistence shown by his side in a victory which kept them on course to win the title.
He added: "The most important thing is the three points. The importance is obvious, we started the game needing six wins and one draw [to be assured of finishing top of the table] and now we need five wins and one draw. It was difficult and deserved.
"We controlled the game and had the ball in the first half, tried to penetrate through a well-organised defence, we try and try and finally scored a goal with a great combination."
