The Welsh Premier League side led at half-time thanks to Wayne Riley's goal.
But after the break Damir Sovsic fired the visitors level, before goals from Mirko Maric and Marko Kolar made it 3-1.
Broughton goalkeeper James Coates saved Sovsic's penalty late on.
This is Airbus' third attempt at progressing beyond the first qualifying round.
Andy Preece's side lost on away goals to Latvian side FK Ventspils in 2013 and were beaten 3-2 on aggregate to Haugesund of Norway last season.
He told BBC Radio Wales he was "proud" of his players.
"We were playing a top team from the Croatian Premier League. I'm really proud," he said.
But the former Bury manager was unhappy at the way his side conceded the equaliser following a counter attack by the Croatians.
He wondered what might have been had Zagreb not scored from that move.
"It was just a lack of discipline. If we'd done our jobs they wouldn't have scored that first goal and from there who knows what might have happened?" pondered Preece.
"Because they [Zagreb] would have had a decision to make if 1-0 was good enough or if they thought they needed to go for an away goal."
The return leg will be played in Croatia on Thursday 9 July.
Croatian side Lokomotiva Zagreb came from behind to beat Airbus UK Broughton 3-1 in the Europa League first  qualifying round, first leg at Bangor's Nantporth.