Batshuayi quickly restored Dortmund's lead just 15 minutes after his introduction and the Belgian netted a last-gasp victor having seen Danny Blum level matters just three minutes earlier in stoppage time.
The draw extends Leipzig's winless run to four league games, and leaves them four points behind the Champions League spots.
Christian Pulisic, restored to Dortmund's starting lineup after being one of the few to impress in the side's Europa League defeat at home to Salzburg, displayed his intent from the start, whipping in a sharp cross that Andre Schuerrle headed wide.
Pulisic had been a sub for BVB's last three league matches dating back to an 85-minute turn on February 10 versus Hamburg.
Kevin-Prince Boateng drew a smart save from Dortmund's Roman Buerki after the break, and the visitors maintained their pressure.
Dortmund saw out the remainder of the half without incident, but the visitors improved in the second half and were asking questions of the home defence just three minutes after the restart.
"I am very relieved because I haven't been scoring recently", said Batshuayi, who had not scored in three league games before Sunday.
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Heading into the break with a 7-0 score after Sammy Oliech booted home the extras was a good result for the Kenyans. Sevuloni Mocenacagi scored two tries, the second of which launched Fiji's second-half surge.
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And the visitors got the leveller their hard work deserved in the 75th minute, substitute Jovic heading home another unsafe De Guzman cross, with Dortmund captain Marcel Schmelzer culpable for losing his man.
Arguably the best chance of an uninspiring game fell to Mario Gomez, who headed wide for hosts Stuttgart with 10 minutes remaining.
But Eintracht switched off, Pulisic and Batshuayi linking up to stunning effect and the on-loan striker smashing in to apparently settle the match.
Not to be upstaged, Batshuayi scored two minutes later to restore the home side's lead after being played through by Pulisic.
Dortmund had not counted on Eintracht's resilience, though, and they levelled again in the first minute of added time, Blum tapping in at the back post from a Danny da Costa cross.
- Marco Russ scored the fifth own goal in his Bundesliga career - in Bundesliga history, only two players scored more: Nikolce Noveski and Manfred Kaltz (6 each). The 19-year-old American now has four goals and five assists this season.
It was the latest Dortmund goal in the Bundesliga since Robert Lewandowski struck against in Mainz in November 2013.