With European champions Bayern at home to strugglers Mainz on Sunday, Leipzig took the chance to go a point ahead as Olmo fired through the legs of Stuttgart goalkeeper Gregor Kobel.
Stuttgart's Swiss shot-stopper had earlier pulled off some heroics by using his foot to save a Emil Forsberg first-half penalty attempt.
Leipzig also wasted a golden chance when Amadou Haidara fired wide just after the break.
With 23 minutes left, Spain midfielder Olmo finally got the breakthrough, finishing a move he started after Leipzig had moved the ball across the Stuttgart penalty area.
Earlier, Bayer Leverkusen blew the chance to take first place after stumbling to a 2-1 defeat at Eintracht Frankfurt which saw them drop to third as the league resumed after a two-week break.
"I have made my feelings clear in the dressing room," Leverkusen head coach Peter Bosz told Sky.
"We deserved to lose. Frankfurt had a better claim to the victory."
Leverkusen midfielder Nadiem Amiri scored with a deft back-heel flick to give the visitors the lead after only 10 minutes in Frankfurt.
However, Amin Younes equalised to make it 1-1 at half-time before Japan midfielder Daichi Kamada put in a cross which Leverkusen defender Edmond Tapsoba turned into his own net.
Before kickoff, Frankfurt director Fredi Bobic confirmed reports his club are interested in Bayern's teenage striker Joshua Zirkzee, after selling Dutch forward Bas Dost to Brugge.
Zirkzee could back up Frankfurt's top scorer Andre Silva, who has notched nine league goals this season.
"He is a good, young footballer with a lot of potential and we are looking for a player of his profile," said Bobic.