French striker Odsonne Edouard gave the visitors a stunning lead in the Red Bull Arena after less than two minutes when he used his strength to fire in from 12 yards.
The Austrian champions, who reached the semi-final of the tournament last season, found Brendan Rodgers’ side resolute in defence until the 55th minute when Israel international Munas Dabbur levelled.
Takumi Minamino put the home side ahead six minutes later before Celtic winger James Forrest was sent off by referee Serhiy Boiko in the 72nd minute for denying Salzburg skipper Andreas Ulmer an obvious goalscoring opportunity.
Dabbur slotted home the resulting penalty to take Salzburg to the top of Group B with six points from two fixtures.
Celtic’s next game is away to German side RB Leipzig on October 25 and, while they will travel with three points from their match day one win over Rosenborg, will surely do so with some trepidation after their second-half display here.
Rodgers was dealt a blow before the game when skipper Scott Brown was forced to pull out with a hamstring strain.
His place was taken by Youssouf Mulumbu, who was making his European debut for the Hoops, while left-back Kieran Tierney captained a side which enjoyed an early boost.
There looked no immediate problem for the hosts when midfielder Callum McGregor helped Tierney’s pass forward but 20-year-old Edouard - the club’s £9million record signing from Paris St Germain this summer - brushed aside the weak challenge of Andre Ramalho and calmly placed his right-footed shot past goalkeeper Alexander Walke.
The goal stung Salzburg and they upped their tempo and forced Celtic back towards keeper Craig Gordon, who in the 12th minute watched a header from Minamino fly over his bar, before Dabbur drove just wide from the edge of the box.
The Hoops rearguard stood firm under relentless pressure, although Minamino again got a header in from a Xaver Schlager cross but was unable to direct it on target.
A minute from the interval Edouard had the ball in the Salzburg net after Walke had parried a Forrest shot but was rightly ruled offside.
Salzburg started the second half at top speed and Gordon did well to block a close-range effort from Dabbur for another corner which was defended.
However, the equaliser arrived 10 minutes after the break when Hannes Wolf collected a Minamino head-flick and cut the ball back for Dabbur to squeeze his shot past Gordon from six yards.
Scott Sinclair replaced Leigh Griffiths on the hour-mark but a minute later the home side went ahead when Ulmer’s cut-back was tucked away by Minamino at the second attempt, the direction of the points now changed completely.
Salzburg swarmed all over the Scottish champions looking for a third which came after Forrest bundled Ulmer to the ground inside the box after being caught in possession, with Dabbur taking care of the spot-kick.
In the 87th minute Salzburg substitute Zlatko Junuzovic struck the post with a drive and Celtic survived on that occasion, but a night which started so promisingly for the Parkhead club ended in disappointment with another tough test in Germany next on the European agenda.