Dinamo Zagreb's 1-0 loss at Astra in the other Group D game meant Ronny Deila's side are uncatchable by both those sides with only a trip to Croatia remaining.
The Scottish champions, however, were in disarray in the early stages against Salzburg as Brazilian striker Alan struck twice in a dramatic first 13 minutes and the visitors might have stretched their lead before Hoops' midfielder Stefan Johansen reduced the leeway with a drive on the half-hour mark.
The home side dominated the second period but could not get back on level terms before Salzburg substitute Naby Keita grabbed a breakaway third in injury time but it mattered not as Dinamo failed to win against the Romania.
Despite the importance of the fixture, the match had failed to capture the imagination of the Hoops fans.
The top tier of the 60,000-capacity stadium remained unopened for the visit of Salzburg whose boss Adi Hutter wanted his team become the first Austrian side to win on Scottish soil.
As expected skipper Scott Brown returned from suspension to take his place in the Celtic midfielder while Efe Ambrose replaced the injured Jason Denayer.
However, Leigh Griffiths was handed his first start since August in the absence of ineligible striker John Guidetti and midfielder Callum McGregor also came in from the cold as James Forrest and Kris Commons dropped to the bench.
Salzburg had their Spanish captain Jonatan Soriano spearheading the attack along with Brazilian striker Alan who stunned the home fans twice in a spectacular opening.
The first goal came when midfielder Massimo Bruno cut in from the left and when his right-footed drive from 25 yards deflected off Charlie Mulgrew, it fell fortuitously to Alan and he quickly fired past the helpless Craig Gordon.
Three minutes later the Hoops keeper made a point-blank save from Salzburg defender Martin Hinteregger's close-range header but the Scotland international had no chance when Andre Ramalho headed Christoph Leitgeb's corner across the face of the goal leaving Alan, again unmarked, to volley in from four yards out.
Deila's side threatened, it seemed, to unravel completely.
McGregor had a penalty appeal waved away by Turkish referee Halis Ozkahya as Celtic tried to fight back but at the other end, Leitgeb forced Gordon into a decent save with a long-distance drive.
There was momentary confusion in the 22nd minute when Soriano fired a 25-yard indirect free-kick past Gordon and into the net but the Scotland international signalled that the ball had not touched anyone on the way in and referee Ozkahya agreed.
However, it was another indication of the danger posed by the slick Austrian side who might have had a penalty when in the 25th minute Alan went to ground inside the Celtic box after a challenge by Hoops' defender Virgil van Dijk.
It was painful viewing for the Celtic fans but the home side revived their fortunes in the 30th minute when Johansen swept an Emilio Izaguirre cross past Peter Gulacsi from just inside the box.
The visitors looked a different team from the one which had had bossed the early part of the match as the home side took control.
Forrest and Commons replaced McGregor and Anthony Stokes respectively in the 67th minute as Celtic looked to capitalise on their increasing pressure but inevitably chances were taken in defence and three minutes later Alan missed out on his hat-trick by a matter of inches from Soriano's cut-back.
In the 74th minute, after Griffiths' right-footed effort from just inside the box missed the target by inches, he had a powerful close-range header brilliantly saved by Gulacsi.
The Glasgow side peppered Salzburg's box with crosses in the final stages but in the three minutes of added time they were undone again when Keita headed in a third on the counter-attack but the result from Romania guaranteed European football at Parkhead after Christmas.
Source : PA
Source: PA