The Spain international, a new arrival from Atletico Madrid after a prolific season in La Liga, got his Chelsea career off to a flying start with a well-taken goal in the second half in Slovenia.
A disjointed Chelsea had fallen behind in the first half after Nik Kapun had lashed a finish beyond young goalkeeper Matej Delac, but Costa had his say soon after the break.
Kurt Zouma then struck the winner for the Premier League outfit, converting from close range after Fernando Torres had seen his shot palmed back into the mass of bodies in the middle of the box.
The hosts hit the front five minutes before the interval, Kapun smashing high into the net after Darijan Matic's chested lay-off on the edge of the area.
Chelsea were level in the 56th minute, though, as Costa ran on to a pass from another new signing - Cesc Fabregas - and thumped a finish high into the net.
And Mourinho's team were in front shortly after, Zouma stabbing the ball home after Seliga had parried Torres' shot back out into the danger area.
There was a moment of controversy in the 66th minute as Chelsea thought they had extended their advantage.
Seliga did brilliantly to tip Zouma's header onto the bar, with Branislav Ivanovic scuffing the rebound goalwards.
The ball appeared to have crossed the line, with the help of John Terry, but the referee ruled the goal out, seemingly for what he felt was a foul by Chelsea's captain on the home goalkeeper.
The Premier League club continue their pre-season preparations against Vitesse on Wednesday.
