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Two goals by Dortmund striker Erling Braut Haaland were bookended by Elvedi netting from set-pieces in the action-packed opening 32 minutes.
Algeria left-back Ramy Bensebaini restored Gladbach's lead early in the second half before substitute Marcus Thuram headed the hosts' fourth.
After home wins over league leaders Bayern and second-placed RB Leipzig already this season, Gladbach claimed another big-name Bundesliga scalp.
It was also a significant win for their coach Marco Rose, who the German media are tipping to replace caretaker boss Edin Terzic at Dortmund next season.
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After 12 straight defeats to Dortmund, Gladbach earned their first win over the Black-and-Yellows since April 2016.
Since their impressive 3-1 win at RB Leipzig at the start January, Dortmund drop to fifth after failing to win their last three games.
In the other direction, Gladbach are up to fourth after taking 13 points from a possible 15 in 2021.
They are eight points behind leaders Bayern, who are at bottom side Schalke on Sunday.
Breel Embolo came on for Gladbach in the second-half despite police claims, which the Swiss striker denies, that he attended a party in Essen last Saturday that broke Germany's lockdown rules.
Dortmund made a lethargic start at Borussia Park and a Gladbach goal inside the opening 60 seconds was ruled out after a lengthy VAR review.
However, the hosts took the lead with 11 minutes gone when Lars Stindl's curling free-kick dropped in behind the defence to be headed home by Elvedi.
Dortmund roared back with two goals inside six minutes, both created by Sancho for Haaland, who fired home his first from a tight angle on 22 minutes.
After Jude Bellingham won the ball back for Dortmund, great footwork by Sancho then again unpicked Gladbach's defence for Haaland to blast his side 2-1 ahead on 28 minutes.
However, another Stindl free-kick on 32 minutes could only be parried by Roman Buerki and Elvedi tapped home the rebound from close range to make it 2-2 at the break.
Gladbach regained the lead five minutes into the second-half when Bensebaini swung in a shot which curled outside Buerki's grasp.
Dortmund conceded their third goal of the night from a set-piece when Thuram headed home a Florian Neuhaus corner 11 minutes from the whistle.
It was the France forward's first appearance after suspension for a spitting at an opponent last month.
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