Mubarak Wakaso scored on his Celtic debut and Scott Brown netted the other goal while the Austrian side twice hit back through Alan and Jonathan Soriano.
The opening goal came on 14 minutes when an Efe Ambrose throw-in on the Celtic right was worked across to the far side of the box by Stefan Scepovic, Brown and Kris Commons to Wakaso, who sent a low curling shot back into the right corner.
Salzburg levelled nine minutes before the break as Kevin Kampl slipped a pass through the centre of the Celtic defence for Alan and his drive from the edge of the box was deflected off Ambrose with Craig Gordon already diving in the opposite direction.
Brown struck on the hour when Callum McGregor got past his man and crossed into the centre of the box where Andre Ramalho headed the ball clear but the volley by the Celtic skipper from almost 30 yards spun off the same player into the corner.
Jonathan Soriano (78) drew the home side level for the second time with a superb dipping free-kick from 25 yards after Emilio Izaguirre was penalised for some sloppy defending.
Three minutes after the opening goal Kampl lifted the ball just wide of the far post as Salzburg worked an opening from a throw in on their right, and Gordon then saved a low shot from Soriano at his near post from a free-kick.
And from another free-kick Gordon produced a wonderful save to turn a Ramalho effort round the post after the Salzburg defender peeled away at the far side of the penalty area to send a header towards the opposite corner.
Celtic were almost caught out again just before the break from a swift counter-attack but Marcel Sabitzer shot wide of Gordon’s right post from 20 yards with Kampl available on the overlap.
Gordon again came to the rescue just after the break when Alan spun just inside the Celtic penalty area and sent a rising shot towards the top corner, but the big keeper pushed the ball over the target.
Scepovic almost caught Peter Gulacsi out with an audacious lob from just inside the Salzbug half but the ball dropped against the side netting with the keeper well beaten.
At the other end Gordon got down to get a hand on Sabitzer’s daisy-cutter and scrambled back on to his feet to turn Alan’s follow up header round the post five minutes before Celtic restored their lead.
Alan shaved the Celtic post with another shot from the edge of the box and Gordon held a free-kick from Sabitzer three minutes before the second equaliser from a similar distance.