Dortmund win top of table clash
The 18-year-old Germany under-21 midfielder showed why he is one of the most promising prospects in the league when he weaved his way into the box and wrong-footed keeper Christian Wetklo with a clever low drive in the 26th minute.
Goetze turned provider in the 67th minute, flicking on a through ball for Paraguay international Lucas Barrios to round Wetklo for his fifth league goal of the season.
Mainz, lacking their usual offensive spark, had Eugen Polanski's penalty saved by keeper Roman Weidenfeller in the 48th minute.
Dortmund equalled a club record with their fifth successive away win. Their fifth straight away victory from the start of the season is also a Bundesliga record.
They are top with 25 points from 10 games, one ahead of Mainz.
Hosts Hoffenheim crushed Hanover 96 4-0 to move up to third place on 18 points, level with Bayer Leverkusen who beat Schalke 04 1-0 on Saturday.
Hanover played most of the match with 10 men after Manuel Schmiedebach was sent off for a second yellow card on 19 minutes.
"We applied our game plan very well against a tough opponent," said Dortmund coach Juergen Klopp, who spent 18 years with Mainz as a player and coach.
"The saved penalty was a key moment. We had luck on our side there. As for the Bundesliga title, we are giving it exactly zero thoughts at the moment."
It may have been a sentimental return to Mainz for Klopp but his team was in no mood for nostalgia.
Dortmund quickly took the initiative and missed three excellent chances after only 20 minutes.
Mainz keeper Wetklo then miskicked a pass from teammate Nikolce Noveski and watched the ball roll just wide of his own goal.
Goetze, a product of Dortmund's youth academy, gave the visitors a deserved lead with a fine solo run, shaking off three defenders before slotting home.
Dortmund did not ease off, hitting the post with Barrios eight minutes later with Mainz's defence in tatters.
The hosts did get a chance to level in the 48th minute but Weidenfeller saved Polanski's penalty.
Wetklo made two spectacular saves to deny Mats Hummels and Shinji Kagawa from close range before Barrios made sure of all three points for Dortmund with a fine finish.