Nik Prelec, who scored once in 24 appearances for then-Championship side Oxford last season, nodded Rangers down into the Conference League 10 minutes into the second half as McInnes' slow start on his return to Glasgow was extended to a fourth game with a 3-2 defeat over two legs.
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The hosts had cancelled out their Polish visitors' first-leg lead inside the opening 20 minutes when former Celtic defender Yuki Kobayashi was penalised by VAR for a handball inside his own box, with Lawrence Shankland's debut goal since his summer move from Hearts bringing Rangers level from the spot.
McInnes's side had more chances to add to their early lead on the night when Thelo Aasgaard fired just wide, while Ryan Naderi also went close before the interval.
But a limp second-half showing was duly punished by Prelec and despite huffing and puffing, the hosts never did enough to test goalkeeper Slawomir Abramowicz after the break, leaving McInnes and his side to take in jeers from around the ground as their Europa League campaign fell at the first hurdle - their earliest exit from the competition since 2017.
The key moments from Ibrox...
- 17: VAR! A VAR review is called after Yuki Kobayashi appears to handball a cross - with the referee pointing to the spot.
- 19: GOAL! Laurence Shankland pulls Rangers level on aggregate from 12 yards.
- 35: CLOSE! Thelo Aasgard strikes from the edge of the box and is inches wide from putting Rangers ahead in the tie.
- 46: SUB! Ross McCrorie replaces Aasgard at the break.
- 55: GOAL! Nik Prelec stuns Ibrox when he nods home at the back post to restore Jagiellonia's aggregate lead.
- FT: BOOS! The full-time whistle is met by a loud and long chorus of boos as Rangers are jeered off following their Europa League exit.
