Serhou Guirassy quickly cancelled out goals from Raphinha and Ferran Torres in the second half in Germany on Wednesday night.
But Torres struck again with five minutes left to secure Barca a fifth win from their six group games and leave them three points behind leaders Liverpool with two matches left.
Dortmund's second defeat - their previous one coming away to Real Madrid - leaves last season's runners-up a point outside the top eight.
Arsenal and Lille are among six teams all on 13 points, two behind Barca and one above Dortmund, after each also winning on Wednesday night.
French side Lille recovered from throwing away a two-goal lead to beat Sturm Graz 3-2 thanks to Hakon Arnar Haraldsson's 81st-minute winner.
And Bukayo Saka's brace and a goal from Kai Havertz helped the Gunners to a comfortable 3-0 win over Monaco.
Bukayo Saka (r) scored two goals for Arsenal against Monaco. (AP PHOTO)
Atletico Madrid extended their winning streak to 10 matches in all competitions by beating Slovan Bratislava 3-1 and boosting their hopes of avoiding a two-legged last-32 tie.
Julian Alvarez and Antoine Griezmann both scored in the first half to put the Spanish side in control.
David Strelec's penalty pulled one back for the Slovakian champions, but Griezmann quickly added his second as Diego Simeone's side recorded their fourth victory from six group games to sit level on points with Dortmund and Bayern Munich.
AC Milan joined the trio on 12 points after Tammy Abraham's 87th-minute winner earned a 2-1 victory over Red Star Belgrade.
Fellow Serie A side Juventus beat Manchester City 2-0 to move on to 11 points.
Second-half goals from Dusan Vlahovic and Weston McKennie left Pep Guardiola's side sitting just one point above 25th-placed Paris St Germain, who face City in their next European game.
Elsewhere, Benfica were held to a 0-0 draw at home to Bologna, Feyenoord beat Sparta Prague 4-2 to move into a play-off place and Stuttgart thrashed bottom club Young Boys 5-1.
In the night's one Europa League match, Inaki Williams scored twice as Athletic Bilbao went top of the table with a 2-0 win at Jose Mourinho's Fenerbahce.
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