Leverkusen staged a clinical chance conversion and shocked the hosts with the opener as Dominik Kohr made the most of Wendell's assist to break the deadlock with 22 minutes played.
The "Werkself" should have doubled the lead two minutes later but Lazat blocked Kevin Volland's shot on target at the last possible moment.
Leverkusen dictated the pace on the pitch, still Mainz levelled the scores out of the blue when Muto capitalized on a cross from Daniel Brosinski to slot home with a sideways scissor-kick just before the break.
The hosts gained momentum after the restart and turned the tides through Abdou Diallo, who headed home Levin Oeztunali's free kick inside the box in the 57th minute.
Heiko Herrlich's men handed over control and got punished in the 71st minute after Mainz' Suat Serder slotted home Oeztunali's second assist of the day to make it 3-1 on the scoreboards.
Leverkusen tried to pull one back but Stefan Kiessling's header missed the target in the dying minutes of the game.
With the result, Mainz reaped their first three points while Leverkusen remain winless three rounds into the Bundesliga season.
Elsewhere, front-runners Borussia Dortmund failed to extend their lead atop the standings as the "BVB" were held to a goalless draw by ten-men Freiburg.
Eintracht Frankfurt wrapped up their first victory owing to Kevin-Prince Boateng's sole goal of the game against Borussia Monchengladbach.
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