After being pushed all the way to a tie-breaker, which he dominated 7-2, in a tight first set, the Scot secured a break early in the second for a 2-0 lead.
Now fully into his stride, Murray began to pummel the German with punishing groundstrokes before finally sealing the double break and the set at 6-2.
The world number three produced some scintillating tennis to break for a third time to go 3-1 ahead in the third set.
As Mayer served to stay in the match at 5-2, Murray chased down a drop shot before flicking a delightful winner without even having to look at where he was aiming.
That helped him on his way to breaking once again to seal a 7-6(1) 6-2 6-2 victory with just under two hours on the clock.