Moutet, who is not averse to trying his luck with the underarm serve, won three and lost three of the points in which he employed the tactic, winning the match 7-6(3), 7-6(3).
He will now get the chance to use the weapon again in his first ATP Tour quarter-final of the year when he faces American Learner Tien on Thursday.
Tien upset top seed Ben Shelton 6-4, 7-6(2), coming back from 4-1 down in the second set to send his compatriot packing.
A day after earning his first Tour-level win since 2021, Australian qualifier Bernard Tomic's run came to an end as he was brushed aside 6-3, 6-3 by Spain's Roberto Bautista Agut.
