Mourinho conceded the champions' title defence was over after Chelsea's 1-0 win over Norwich on Saturday, just a fourth win from 13 games this season, but reckons Champions League qualification is achievable.
"It's a bit of a relief," said Mourinho, after Costa's first goal since the win over Aston Villa ended a three-match losing streak.
"The pressure was there - I was feeling, the players were feeling too.
"We coped well with that and we got a result which obviously we needed very, very much.
"Fourth position for me is not an impossible mission
If you ask me the title I would say impossible mission.
"Maybe Tom Cruise can do it (but) it's complicated because you have to recover points from four candidates.
"But to grab one of the (teams) that go up and will also have a little bit of a collapse..
the fourth position is a difficult position but a possible target."
Costa admitted earlier this season to reporting for pre-season overweight and the striker, prolific last term in Chelsea's title triumph, had scored just three times prior to the international break.
The Brazil-born Spain striker had spurned two good opportunities and been absent when Eden Hazard fashioned two more before pouncing following Cesc Fabregas' quick free-kick.
Mourinho added: "You don't score goals, you get heavier
Every game that you don't score goals, you get 5kg more
You get heavy and the pressure is there.
"In the first half he missed two chances
The second one, in the last minute, is really a big one
So it was important for him
Important for us, the result and the goal, but I think also for him.
"He's working well, he's a happy guy, he tries everything
He's positive, so if I had to choose somebody to score the winning goal, I would go exactly with him."
Costa's fluency was not what it should be, says Mourinho, who had to remind his striker to return to position quickly moments prior to the goal.
"When you lose confidence, you lose this fluent game
Yes, he can do much better, but again, yes, one goal is very important," Mourinho added.
"I was telling him that he was being caught offside a couple of times, also in the first half, because he was late to recover position.
"When the team tries to reach that, he needs to be in a position to attack that, not in an offside position.
"The goal was a little bit like that, because he was in a position that he could attack space and not in a space offside."
Hazard also showed glimpses of a return to form.
"Every top player that is not performing well they feel it probably more than the others that are not so good," Mourinho said.
"Eden created chances for us, which is important and I think also for his confidence."
Mourinho felt Chelsea should have been well clear in the closing stages but was relieved they claimed just a third clean sheet of the season.
"We should score three, four, five goals, and we didn't," he said.
"You get a little bit disappointed with the way things go
After the goal you had (Kurt) Zouma hit the post
If he doesn't touch the ball it's a goal.
"Then in the last four minutes you are there and you think maybe we score in our own goal in the last minute."
Norwich boss Alex Neil pondered what might have been, too, after feeling aggrieved not to be awarded a penalty when Willian clumsily challenged Robbie Brady.
Neil said: "I generally don't say anything about referees because they have a hard enough job but it is getting extremely frustrating.
"When you are one of the smaller clubs it becomes extremely difficult to accept because those minor decisions mean a great deal to us as we might not have five or six opportunities to score.
"So when you do feel you deserve a penalty you would like it to be awarded
That is where the frustration lies."
Neil felt his side were culpable in Costa's winner.
"That little lapse in concentration ultimately cost us the match," he said.
Source : PA