Zverev flexed his dominant serve, blasting 21 aces and winning 79% of his first serve points on the Grandstand.
Unseeded Marterer, the "lucky loser" that entered the draw after Emil Ruusuvuori withdrew last week, captured a close second set but was ultimately overpowered by the world No. 4.
Zverev looked sharp in the first set, breaking his opponent's serve in the opening game before Marterer battled back to claim a back-and-forth second set in a tiebreak.
The big-serving Zverev looked frustrated at times, tossing his racket at the start of the third set after committing one of 44 unforced errors.
The win sets up a second round clash with the winner of the match between Frenchman Alexandre Muller and Australia's Adam Walton later on Monday.
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