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Murray, who was dumped out in the first round of last week's Open Sud de France, hit 33 winners on his way to a 2-6 7-6 (7-2) 6-3 triumph – his first tour-level victory since the 2020 US Open.
The Scot got off to a slow start and found himself 4-1 down in the first set, with home favourite Haase comfortably seeing things out from there in the opener.
VIDEO: Murray ready to start winning again
Murray needed a tie-break to take the second set before finding himself 3-0 down in the decider. However, he put together a stunning six-game run to claim a fifth career victory over Haas in six meetings.
Next up will be either number four seed Andrey Rublev or qualifier Marcos Giron in the second round.
Elsewhere, Kei Nishikori ended a run of four straight defeats with a 7-6 (7-4) 6-1 victory over seventh seed Felix Auger-Aliassime to set up a clash against Alex de Minaur or John Millman.
"It was a very tight game, he was serving great especially in the first set and he won a lot on first serve," Nishikori said. "I played solidly in the tie-break and served well on the last couple of points. It was easier in the second set, with his injury, but I was more comfortable and striking the ball better."
Murray's compatriot Cameron Norrie, meanwhile, cruised past Georgia's Nikoloz Basilashvili 6-0 6-3 and will play Stan Wawrinka or Karen Khachanov in the next round.
