Daniel Chima's first-half goal means Sligo face an uphill battle to keep their Champions League dream alive at Molde's expense.
The Nigerian struck three minutes before half-time at The Showgrounds as Ole Gunnar Solskjaer's men earned an advantage to take back to Norway for Tuesday's return leg.
Ian Baraclough's men were disappointing in the first half and their only noteworthy effort on goal was a 24th-minute shot from Evan McMillan, which was saved.
Jo Inge Berget hit the woodwork seven minutes before half-time but the hosts did not heed the warning and fell behind four minutes later when Chima broke through.
The 2012 Irish League champions pressed hard for an equaliser in the final quarter, with Joseph Ndo, Jeff Henderson, Lee Lynch, Kieran Djilali and Alan Keane all threatening.
However, Molde maintained their advantage and are now firm favourites to reach the third qualifying round.