They moved onto 54 points from 26 matches, one point ahead of Bayern Munich who host Freiburg on Saturday.
Schalke coach Felix Magath has played down his team's title chances but their climb to the top comes at the same stage as when his former side VfL Wolfsburg's took over the lead last term after 26 games and stormed to their first title.
"My team played like always. More controlled in the first half and more energetic in the second," Magath told reporters.
"But we can thank (goalkeeper) Manuel Neuer for not conceding a first half goal. After we took the lead we failed to put more pressure on. We were lucky to get away with a win."
The former Germany striker, battling for a recall ahead of this year's World Cup, joins Bayer Leverkusen's Stefan Kiessling as top scorer in the Bundesliga.
Edu could have added a second goal with eight minutes left and Stuttgart's defence in disarray when Ivan Rakitic put him through but his powerful low drive flew just wide.
The win is a perfect start to what Magath has described as "the weeks of truth", with matches against Hamburg SV, Bayer Leverkusen and Bayern Munich to follow.
