Just two days after he won the Shenzhen Open for his first title since Wimbledon 2013, Murray had to come through three sets to beat his Polish opponent.
Murray's efforts on Sunday appeared to be telling when he fell 3-0 and then 5-1 down in the first set before he rallied to force a tie-break.
Both men squandered two set points in the breaker before Janowicz clinched it at the third try 11-9.
Murray wasn't broken for the rest of the match, snapping Janowicz's serve once in the second set to win it 6-4 before two more breaks in the decider as he came through 6-2 in two and-a-half hours.
Awaiting the Scot in round two is Uruguay's Pablo Cuevas.