Dortmund thought they had done enough when Batshuayi, on loan from Chelsea, put them 2-1 ahead in the 77th but Danny Blum levelled in the first minute of stoppage time before Batshuayi grabbed his second goal just before the final whistle.
The result lifted Dortmund to 45 points, one behind second-placed Schalke 04, one ahead of fourth-placed Bayer Leverkusen and three in front of Eintracht who have slipped to fifth. Bayern Munich lead the title race on 66 points.
Eintracht's Marco Russ had slid in for an own goal to put Dortmund ahead in the 12th minute as the hosts set a frantic early pace and put the visitors on the defensive.
Dortmund gradually eased off and allowed Eintracht to equalise through Luka Jovic's glancing header in the 75th.
But the visiting side's joy was short-lived as Belgium international Batshuayi, who came on after an hour, fired in from 15 metres, following fine early work from Christian Pulisic, to complete a quick break two minutes later.
Batshuayi, who joined Dortmund in the January transfer window, was relieved to net his first goal after five matches in all competitions without scoring.
He got a second deep into stoppage time to secure three points for Dortmund and cancel out Blum's late equaliser.
RB Leipzig, last season's runners-up, suffered a setback to their hopes of a top-four finish and Champions League qualification after being held to a 0-0 draw at VfB Stuttgart that left them in sixth place on 40 points.
(Reporting by Karolos Grohmann; Editing by Ken Ferris)
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