McInnes made five changes from their Premiership draw with Dundee United and they made the dream start in Poland as Guilherme Montoia's own goal put them 1-0 up.
But Manny Fernandez returned the favour when he turned Kajetan Szmyt's cross into his own net before the Jagiellonia midfielder took advantage of some poor Rangers defending to fire in a second before half-time.
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McInnes brought on Cammy Devlin, Nico Raskin and Findlay Curtis for the second half. For all they improved the side's attacking threat, they failed to make it count.
Jesus Imaz missed a brilliant chance to extend the hosts' lead, with Gers substitute Lawrence Shankland failing to poke home a loose ball to level the match before Curtis' stoppage-time effort crashed off the post.
Rangers have it all to do at Ibrox next week. The winners will face Larne or Iberia 1999 in the Europa League play-off round, with the losing side dropping into the Conference League play-offs.
More to follow...
The key moments from the Chorten Arena
4: GOAL! Rommens sends in a low cross that is just out of Chermiti's reach, but it comes off Montoia and goes in.
13: GOAL! Disaster for Rangers as Szmyt's ball from out wide comes off Fernandez and ends up in the back of the net.
21: GOAL! Szmyt fires in on the rebound after Pandur denies Imaz.
61: CLOSE! Raskin with a brilliant ball to Curtis, but he can't get his effort on target.
68: LET OFF FOR RANGERS! Imaz is through but somehow misses the target.
86: BIG CHANCE! Shankland blocks Fernandez's header from the corner but fails to poke home the loose ball.
