The Spain international appeared to be fouled by Tom Heaton in the first half, having burst into the box, but referee Michael Oliver booked the striker, much to Mourinho's chagrin.
The Portuguese was left frustrated by the match officials after the game and praised Costa's attitude throughout the game, with the €40 million signing from Atletico Madrid equalising for the Blues following Scott Arfield's stunning opener.
"Costa last season was fantastic and in the Spanish league he got five, six yellow cards," the Portuguese told reporters. "We explained what we want in the Premier League, what they like and don't like, he presented himself here today with a fantastic attitude.
"It was a clear penalty and I hope that he doesn't get more unfair decisions. Unfortunately the linesman didn't support him."
Costa scored 36 goals in all competitions for Atletico last season.