Inter narrowly avoid early Europa League exit
Inter's problems began in the 23rd minute of the third qualifying round second leg tie when Hajduk forward Ivan Vukovic was fouled by Walter Samuel and Ante Vukusic converted the penalty.
Inter had a flurry of chances just before the break but fell further behind when Vukovic rifled the ball home from just inside the area in the 68th minute.
Vukusic nearly sent the game to extra time when his low shot was just wide of the post as Inter fans watched in stunned silence.
"It's only August 9, these things happen," said Inter coach Andrea Stramaccioni. "The important thing is that we got through."
Athletic Bilbao, last season's runners-up, also qualified despite losing to Croatian opponents as they went down 2-1 at Slaven Belupo Koprivnica but won 4-3 on aggregate.
Liverpool had no such problems as a 3-0 win at home to Gomel sent them through 4-0 on aggregate.
Fabio Borini and Steven Gerrard put them two goals ahead in the first half before Glen Johnson completed the scoring in new manager Brendan Rodgers' first home game in charge.
Other teams to progress in Thursday's 29 matches included Steaua Bucharest, Dynamo Moscow, Olympique Marseille and APOEL Nicosia, last season's surprise Champions League quarter-finalists.