In heavy snow, David Degan gave Young Boys the lead after 35 minutes before Pavel Pogrebnyak equalised in the 48th minute and substitute Sven Schipplock gave Stuttgart the lead in the 68th minute.
Scott Sutter levelled the game at 2-2 in the 78th minute before Zambia international Mayuka struck in the 81st and 82nd minutes for the Swiss victory.
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With one match remaining in Group H, both Stuttgart and Young Boys have guaranteed spots in the last 32. Stuttgart are atop the group with 12 points with Young Boys second on nine.
With Young Boys' David and Stuttgart's Philipp Degan both starting, the Swiss duo became the first twin brothers to face off in an UEFA competition.
The start of the match at Berne's Stade de Suisse stadium was delayed for 30 minutes due to heavy snow.
Young Boys nearly grabbed the lead just two minutes later as Senad Lulic struck a blast from the edge of the area which hammered off the crossbar.
With heavy snow continuing to fall, referee Alon Yefet ordered the lines of the pitch to be cleared after 29 minutes.
The cleared lines were not needed to see that David Degan gave Young Boys the lead after 35 minutes, turning on a ball just outside the area and shooting a looping ball over Stuttgart keeper Marc Ziegler.
Stuttgart equalised three minutes after the break following a defensive mix-up between home defender Alain Nef and keeper Marco Wolfli. Pogrebnyak promptly pounced on the loose ball to make it 1-1.
Stuttgart took the lead in the 68th minute as Mauro Camoranesi sent a looping ball into the box to Schipplock. The substitute, on for just two minutes, slotted home first time.
The hosts equalised with 12 minutes left as Sutter's blast deflected off Stuttgart defender Georg Niedermeier.
Young Boys took the lead in the 81st minute with Mayuka running onto a throw-in before slotting past Ziegler inside the far right post.
And Mayuka struck just a minute later as Ziegler could not clear Sutter's blast which went to the Zambian, who poked it home.
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Source: DSG
