The Kazakh Premier League champions had already come through three qualifying rounds and go into the hat for Thursday’s draw for the Champions League proper for the first time.
There they can hope to rub shoulders with the likes of superpowers Paris Saint-Germain and Real Madrid.
Celtic meanwhile must console themselves with a place in the Europa League.
With neither side able to score in 210 minutes of football across the two legs, it came down to penalties which Kairat won 3-2.
Celtic, the 1967 European Cup winners, will now drop into the Europa League and face a 5,600 kilometre trip home from Kazakhstan ruing two dreadful 0-0 stalemates and a badly fluffed shoot-out.
“It’s hugely disappointing and I really feel for the supporters who have travelled all the way to be here, the effort and commitment they’ve shown to be here,” said Celtic manager Brendan Rodgers. “We’re bitterly disappointed for them as well as ourselves.”
Japan forward Daizen Maeda missed the decisive penalty after Celtic had been left reeling when Kairat’s 21-year-old keeper Temirlan Anarbekov saved Adam Idah’s opening effort and then Luke McCowan’s weak second one.
Celtic boss Rodgers blamed the quality of the pitch for his team’s failure to score, while the home side were cheered on in an electric atmosphere.
“The pitch made it awful. We were defensively okay but couldn’t quite make the breakthrough,” Rodgers said.
Arne Engels and Callum McGregor converted their penalties but to no avail.
“They made it stuffy and credit to them, over the two games they defended really well,” said Celtic captain McGregor. “Over the two legs we didn’t do enough to hurt them.”
The game was locked at 0-0 after last week’s goalless draw at Celtic Park and the two sides played out a second 0-0 stalemate, despite a late flurry from the visitors in extra-time.
Maeda was though on goal as full-time approached but sent a shot over the bar with just the keeper to beat.
“We had good chances, especially in the second half,” Rodgers said.
“When it goes to penalties, anything can happen.”
Bodo Glimt of Norway beat Sturm Graz 6-2 on aggregate despite a 2-1 defeat Tuesday, while Pafos of Cyprus beat Red Star Belgrade 2-1 by holding the Serbians to a 1-1 draw.
The winners of all seven play-off ties advance to the league phase of the Champions League while the defeated sides switch to the league phase of the Europa League.