Murray, the world No.3, needed only one hour and 37 minutes to dispose of Haase 6-3 6-1 6-3.
The British star was the runner-up at Melbourne Park in 2010 and 2011.
Murray, 25, finally claimed his first major title last year at Flushing Meadows, having lost his first four grand slam deciders.
Murray will play Joao Sousa from Portugal in the second round.
Sousa also advanced to the second round with a minimum of fuss, beating Australian wildcard entrant John-Patrick Smith 6-4 6-1 6-4 on Tuesday.