Murray, the doubles champion at Queen's a month ago, came through his first Grand Slam match since undergoing surgery in January in the first round of the doubles draw but, despite winning the first set, was unable to replicate that success.
The Croatian pair advance to the third round with a 6-7(4), 6-4, 6-2, 6-3 victory.
It took a tie break to separate the two doubles teams but the Croats earned the first break of the match to win the second set.
In the third, Murray sent a backhand long before being broken in a marathon game. One break was all it took to separate the respective teams in the fourth set.
Murray will now turn his attentions to the mixed doubles draw as he and Serena Williams kick off their campaign against Alexa Guarachi and Andreas Mies.