Saul Niguez, Diego Costa and Koke put Diego Simeone’s men on course for the quarter-finals – despite the two-time Europa League champions being below their best.
Saul’s stunning top-corner opener contrasted with the rest of a muted first half.
Then Costa tucked in a second-half rebound after Brazilian-born Russia keeper Guilherme Marinato kept out Antoine Griezmann’s effort.
Finally, Koke planted home Juanfran’s last-minute cutback to leave the Spaniards strong favourites for the second leg against the Russian Premier League leaders.
Borussia Dortmund 1 RB Salzburg 2
Valon Berisha’s double sent Peter Stoger to a shock first home loss as Dortmundboss – as the Austrian visitors took their unbeaten away run in Europe to 10 games.
Salzburg edged a goalless opening half before Berisha drilled home a penalty after Omer Toprak fouled Hwang Hee-chan.
Midfielder Berisha made it 2-0 on 56 minutes with a fierce first-time drive from Stefan Lainer’s centre.
Former Chelsea wideman Andre Schurrle soon poached third-placed Bundesliga side Dortmund back into contention from close range.
But Salzburg held on to stay in a strong position to reach the last eight.