The striker, who moved to Stamford Bridge from Atletico Madrid during the summer, took his tally to seven goals from four Premier League games with a hat-trick in Saturday's 4-2 victory over the Swans.
Mourinho has revealed that he considered resting Costa for the game, but the Spain international, who had been suffering with a hamstring issue, was keen to play.
"The most important thing of a player's character is what affects his performance as a player," Soccerway quotes Mourinho as saying.
"Diego was a risk and a doubt before Everton, a risk and a doubt before Spain, a risk and a doubt before this Swansea game and he played all three games. I tried to rest him, I tried to be sure, to be 100 per cent sure, but he goes on.
"He is a fantastic character. If the injury is a clear injury of course he will not play, but when the doubt is there and he has not the best feelings, but he feels he can play."
Chelsea face Schalke 04 in the Champions League on Wednesday.