Okazaki then equalled his tally for the whole of last season as a Stuttgart player to put Mainz in front again before Muller got his second of the game.
In the day's other game, Schalke and Hamburg shared six goals in a rollercoaster 3-3 draw at the Arena AufSchalke.
Klaas-Jan Huntelaar gave the home side the lead in the second minute, but a Rafael van der Vaart penalty brought Hamburg level 10 minutes later.
An own goal from Timo Hildebrand gave Hamburg the lead in the 24th minute, but Huntelaar's second of the game on the stroke of half-time brought the Royal Blues back on terms.
Lasse Sobiech put Hamburg in front for the second time early in the second half, but Adam Szalai salvaged Schalke a point in an entertaining encounter.