Half-time substitute Kelechi Iheanacho fired Celtic ahead in Belgrade before the Scottish champions had to settle for a point against Red Star in their Europa League opener.
After a first half which saw Republic of Ireland international Liam Scales produce several important interventions to keep the hosts at bay, Iheanacho finished brilliantly in the 55th minute as Celtic made a flying start to the second half.
Benjamin Nygren made a direct run at the heart of the home defence and zipped in a pass to Iheanacho, who took a touch and sidefooted the ball into the top corner from 16 yards.
But they were pegged back by former West Ham, Stoke and Inter Milan forward Marko Arnautovic, who scrambled the ball home following a corner.
Brendan Rodgers' side could not get back on the front foot and would have been more than satisfied to take a 1-1 draw back from Serbia given the overall balance of play in the Rajko Mitic Stadium.
Republic of Ireland striker Evan Ferguson was a second-half substitute for AS Roma as the Italian giants opened their campaign with a 2-1 win at Ligue 1 side Nice.
Ferguson, who joined Roma on loan from Brighton this summer, has started three of their opening four Serie A fixtures this season, including Sunday's Rome derby victory over Lazio.
However, the 20-year-old Meathman was on the bench with Ukraine international Artem Dobvyk starting up front in his stead, before coming on for the final 21 minutes at the Allianz Riviera stadium in southern France.
Before his introduction, Ferguson's namesake Evan Ndicka opened the scoring on 52 minutes before the Ivorian's centre-back partner Gianluigi Mancini doubled the lead three minutes later.
Terem Moffi pulled a goal back for Nice from the penalty spot with 13 minutes to go but Roma held on to claim all three points.
Nottingham Forest's first European match in nearly 30 years ended in a 2-2 draw as Manchester United flop Antony's late strike for hosts Real Betis extended boss Ange Postecoglou's wait for his maiden victory in charge.
The two-time European Cup winners are back in continental competition for the first time since the 1995/96 season and kicked off the league phase with their toughest-looking assignment.
Forest showed spirit and skill in Seville, where Igor Jesus' brace put them on course for a memorable victory as the visitors reacted impressively to Cedric Bakambu's 15th-minute opener.
The Brazil striker turned in Morgan Gibbs-White's low ball in the 18th minute and sparked further bedlam in the away end five minutes later when heading home a Douglas Luiz corner.
Forest should have gone further ahead before the break and then held firm amid building Betis pressure until five minutes from time, when Antony followed up his earlier assist by levelling for last season's Conference League runners-up.
It was a late gut punch that denied Postecoglou a first victory since succeeding popular Nuno Espirito Santo, but the Australian will take heart from the opening game of a competition he won with Tottenham 18 weeks ago.
Elsewhere, Sturm Graz goalkeeper Oliver Christensen fumbled a Franculino Dju corner into his own net just seven minutes into his side's 2-0 defeat to Midtjylland.
Dinamo Zagreb were 3-1 winners at home to Fenerbahce, Feyenoord lost 1-0 at Braga and Freiburg beat Basel 2-1.
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