Freiburg also wasted the chance to close in on a top-four place as Union Berlin all-but secured Bundesliga safety with a 2-1 away win.
Off the field RB Leipzig have sacked coach Marco Rose during a disappointing Bundesliga campaign and appointed Zsolt Low until the end of the season.
Mainz, who have never been in Europe's top competition, remain fourth but failed to increase the gap on the chasing pack as their goal difference suffered. They had not lost in six games but now hold just a two-point cushion over fifth-placed Borussia Mönchengladbach.
Dortmund, in 10th, moved to within four points of a guaranteed European berth with seven games to go.
Maximilian Beier, up front in place of the absent Serhou Guirassy, gave Dortmund the lead with a tap-in and Emre Can headed in a corner just before the break. Germany under-21 star Beier matched Can's trick midway through the second half before Paul Nebel's consolation.
"We let in two stupid goals from set pieces. We deserved to lose," Mainz striker Jonathan Burkardt told DAZN. "It hurts when you concede two easy goals."
Freiburg stay seventh, level on points with RB Leipzig and three points behind Mainz in the final Champions League spot. Leaders Bayern Munich, second-placed Bayer Leverkusen and Eintracht Frankfurt, in third, all won earlier this weekend.
Union, meanwhile, climbed eight points clear of the relegation play-off spot.
Lucas Höler opened the scoring for the hosts on 29 minutes when he finished off a fine team move, but Rani Khedira levelled almost immediately for Union with a shot from the edge of the area.
Andrej Ilic then timed his run perfectly to head home on 48 minutes to stun the home fans.
Rose's exit follows Leipzig's defeat to Monchengladbach on Saturday and ahead of their German Cup semi-final on Wednesday at VfB Stuttgart.
"We've still got the chance to achieve our goals in two competitions," Low said. The former Hungarian defender has worked as an assistant coach at Leipzig under Ralph Hasenhuettl and Ralf Rangnick.
Rose, 48, had been in charge of Leipzig since late 2022 and guided them to the German Cup title in 2023.
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