PSG host relegation battling Le Havre on Saturday
The Norwegian 19-year-old, who has already netted eight times in five Bundesliga matches, stabbed in from close range after a quickfire passing move in the 69th minute and then thundered in an unstoppable shot for his second in the 77th, two minutes after Neymar had levelled for the French champions.
Haaland has now scored 10 Champions League goals in seven games this season -- eight for Salzburg in the group stage.
Neymar, who had been out for weeks with a rib injury, hit the post late on. The return leg is in Paris on March 11.