Klaas-Jan Huntelaar hit a brace for Schalke, the first after only 80 seconds, while Adam Szalai scored on his debut as the Royal Blues twice came from behind against Hamburg.
Forward Shinji Okazaki scored on his league debut for Mainz 05 against his previous club VfB Stuttgart to give his new side a 3-2 win.
Overall, the weekend's nine games produced 37 goals with only Augsburg, VfL Wolfsburg and promoted Eintracht Braunschweig failing to score.
Huntelaar got Schalke off to a flying start when he raced through the Hamburg defence and planted a right-foot shot past Rene Adler after less than two minutes at the Veltins-Arena.
Fellow Dutchman Rafael van der Vaart quickly equalised with a 12th minute penalty, Maximilian Beister headed the visitors in front and the drama continued as Huntelaar headed Schalke level with the last touch of the first half.
There was no let up after the break as Hamburg went back in front with a Lasse Sobiech header, only for Szalai to complete the scoring when he snapped up a rebound following a mistake by Hamburg goalkeeper Adler in the 72nd minute.
Minutes earlier, Adler had been Hamburg's hero with a brilliant point-blank save to keep out a Szalai header.
Joel Matip had two good chances to win the game for Schalke, who are hoping to better last season's fourth-placed finish.
In the earlier game, Nicolai Mueller opening the scoring in the 14th minute only for Vedad Ibisevic to equalise two minutes later.
There was a suspicion of offside when Okazaki put Mainz back in front in the 65th minute before Mueller slipped between two defenders and poked his shot past Sven Ulreich in the 78th.
Japan forward Okazaki, one of four Mainz debutants, left Stuttgart six weeks ago to join Mainz as a replacement for Szalai, their leading scorer last season.
(Writing by Brian Homewood in Berne; Editing by Clare Fallon/Rex Gowar)
