Leipzig fought back with second-half goals by Christopher Nkunku and Yussuf Poulsen before Sorloth's winning header in the 93rd minute after Gladbach had led 2-0 at half-time.
Sargent scored one goal, had another ruled out, and saw another goal he set up ruled out on a busy night for the video referee.
Bundesliga: Erling Haaland spares Borussia Dortmund blushes in Hoffenheim draw; Bayer Leverkusen drop points
Erling Braut Haaland's 81st minute strike averted a fourth defeat in five Bundesliga games for crisis-hit Borussia Dortmund.
Bundesliga: Eintracht Frankfurt back to fourth spot with Hoffenheim win; Werder Bremen-Arminia Bielefeld match postponed
Filip Kostic steered Eintracht Frankfurt back to fourth in the Bundesliga with a 3-1 win at Hoffenheim on Sunday, as the day's late game was called off due to a heavy snowstorm sweeping across Germany.
Rot-Weiss Essen struck twice in extra time to knock Leverkusen out with a 2-1 defeat.
Bayern Munich’s closest challengers, RB Leipzig and Bayer Leverkusen, both lost in the Bundesliga on Saturday to give the eight-time defending champions a chance to move seven points clear at the top.
The victory was Leverkusen's first in five league games and puts them second on goal difference, level on 32 points with RB Leipzig, who host Union Berlin on Wednesday.
For the second consecutive Saturday, Leipzig, who lost 3-1 last weekend at home to Borussia Dortmund, missed out on first place.
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.
Bundesliga: Edin Terzić makes winning start as Borussia Dortmund head coach with win over Werder Bremen
Dortmund captain Marco Reus scored the winner from close range on a rebound in the 79th minute after Bremen goalkeeper Jiri Pavlenka saved his penalty attempt.
Bundesliga: Bayern Munich remain on top after draw against Union Berlin: Borussia Dortmund lose to Stuttgart at home
The victory left Leipzig a point clear of defending champions Bayern, who can regain first place at FC Union later on Saturday in Berlin.
Bundesliga: Unbeaten Bayer Leverkusen swat aside Schalke to climb second; Stuttgart beat Werder Bremen
Schalke have not won since January, and are now just five matches away from equalling the unwanted record set by Tasmania Berlin in 1966.
Bundesliga: Wout Weghorst stars with two goals as Wolfsburg beat Werder Bremen 5-3 after Maradona tribute
VfL Wolfsburg and Werder Bremen played an eight-goal thriller as Wolfsburg won their Bundesliga game 5-3 after paying tribute to Diego Maradona.
Bundesliga: Erling Haaland nets four as Borussia Dortmund rout Hertha Berlin; Bayern Munich draw at home
Erling Braut Haaland netted four goals and Youssoufa Moukoko became the youngest player in Bundesliga history on Saturday as Borussia Dortmund routed Hertha Berlin 5-2 to trim Bayern Munich's lead at the top of the table to just a point.
Dortmund had three times as many shots on goal and dominated with 80 percent possession, which left their Swiss coach Lucian Favre to admit, "we are very disappointed, it's hard to accept this defeat".
Werder Bremen hung on to seal Bundesliga survival on away goals by drawing at second-division Heidenheim 2-2 in the second leg of their relegation-promotion playoff on Monday.
Bundesliga working with German government to bring back fans into stadiums, says league's CEO Christian Seifert
Bundesliga is working with the government on a plan to get fans back into stadiums during the coronavirus pandemic, their CEO says.
Werder Bremen, who finished third from bottom in the top division, will have to do without captain Niklas Moisander for the second leg on Monday after he was sent off late with his second yellow card for a poor challenge.
Borussia Monchangladbach secured the fourth spot ahead of Bayer Leverkusen on the final day of the Bundesliga season with a 2-1 win over Hertha Berlin.
Bundesliga: Robert Lewandowski breaks goalscoring record; Erling Haaland brace helps Dortmund secure second place
It means Dortmund are assured of second place with one game of the season remaining. Despite defeat, Leipzig are virtually assured of Champions League football next season.