The tie was in the balance after a goalless first leg in Glasgow last week but that changed inside a minute when the Hoops defender brought down Guilherme Afonso inside the box.
After Majstorovic had walked, Pascal Feindouno scored from the spot and from then on Celtic were fighting an uphill battle they never looked like winning.
Feindouno added a second after the break with Celtic defender Charlie Mulgrew and Giovanni Sio swapping goals late in the game as the home side ran out 3-1 aggregate winners.
Celtic have now won only one out of their last 30 games away from home in Europe and did not give themselves much of a chance in this one, but nevertheless the spotlight will swing onto Neil Lennon who admitted his credibility was on the line ahead of this game.
In his first European foray as permanent boss last season the Irishman's side lost to Braga and FC Utrecht in their respective Champions League and Europa League qualifiers but no doubt he will point the finger at Majstorovic for his contribution.
Lennon's team selection had been bold.
Beram Kayal, Kelvin Wilson and Gary Hooper all returned after recovering from injury while Cha Du-ri came in for Adam Matthews at right-back and striker Georgios Samaras replaced top scorer Anthony Stokes, which raised a few eyebrows.
The Irishman dropped to the bench along with dropped Kris Commons with Shaun Maloney and Victor Wanyama left out altogether. But it was a horror start for Celtic as they found themselves a goal down and a man down within three minutes.
There was less than a minute played when Majstorovic was caught out of position from what looked like a straightforward long ball from Geoffrey Serey Die and felled Afonso inside the box as he raced in on goal.
After referee David Fernandez Borbalan showed the Hoops defender the red card, playmaker Feindouno, who had previously been on trial at Parkhead, dispatched the ball past Fraser Forster with some confidence.
In the reshuffle, Mulgrew moved in to the centre of defence with Joe Ledley going to left-back and Samaras dropping deeper as the injured Afonso had to be replaced by Aleksander Prijovic.
It was an uphill task for the visitors thereafter. Mulgrew put a good block in on substitute Prijovic as the Hoops defence was exposed again with a long ball.
In the 27th minute Forster had to dive low to his left to beat out a Feindouno free-kick after Kayal was penalised for hand ball 25 yards out.
The game took a bit of a breather but Cha had to put in a crucial saving block on Feindouno after the Sion player had brilliantly made a yard of space for himself in the Celtic box.
Sion broke with pace again in the 38th minute following a Hoops corner with Kayal lying on the ground following a challenge by Prijovic in the middle of the park.
The Sion substitute ran free on to a Feindouno pass but he never looked confident and his low drive skipped past Forster's left-hand post.
Two minutes from the break, after Sion defender Michael Dingsdag was booked for a foul on Samaras 25 yards out, Mulgrew's curling free-kick was heading inside the post before it was palmed away to safety by Andris Vanins.
It was not an encouraging start to the second half for the Parkhead men.
In the 51st minute their defence was cut open all too easily again when Sio played in Prijovic but Forster got a touch to his drive from 12 yards and ball fortuitously came back off the post.
Moments later, as Sion moved effortlessly into the Parkhead box once more, Forster had to make a save at the feet of Sio who, in truth, dallied too long on the ball.
All the while the Hoops knew just one goal would win them the tie but that looked unlikely in the 63rd minute after Ki Sung-yueng had fouled Sio 30 yards from goal.
Feindouno looked as if he made a mess of it when his initial effort was blocked but he was allowed another go and with the Celtic defence static he drove the ball low past Forster who stood and watched the ball nestle in the corner of the net.
It was all but over for Celtic. In the 69th minute Hooper had the ball in the net from a Ki cross but was ruled offside.
Mulgrew gave Celtic some hope when he drilled in a free-kick in the 77th minute after Hooper had won a foul 20 yards out.
But in the 83rd minute, with Celtic pushing for the equaliser, Sio raced away from Ki over the halfway line and cheekily dinked the ball over Forster to seal the Swiss side's place in the group stages.