Reigning champions Barcelona were held to a 1-1 draw at Roma while Chelsea thumped Maccabi Tel-Aviv 4-0 on Wednesday.
An Alex Oxlade-Chamberlain own goal and a header from Junior Fernandes gave Dinamo three points as France forward Olivier Giroud was dismissed for two first-half bookings for Arsenal.
Zagreb took the lead from virtually their first attack of the game as Leonardo Sigali slipped in Josip Pivaric whose shot was saved by David Ospina, only to rebound off Oxlade-Chamberlain and go in.
Dinamo capitalised on 58 minutes as Fernandes headed home a Paulo Machado corner at the near post as Arsenal's defence went walkabout.
Theo Walcott gave the Gunners hope 11 minutes from time with a cool, low finish from Alexis Sanchez's pass, but it was too little too late as Dinamo earnt their first ever win over an English side.
In Rome, Barcelona opened the scoring on 21 minutes as Uruguay forward Luis Suarez headed home at the back post from a yard out after he was found by Ivan Rakitic.
Chelsea finally kick-started their season as Jose Mourinho's outfit cruised to victory.
The Blues could even manage to miss a first half penalty through Edin Hazard before Brazilians Willian and Oscar, and Spaniards Diego Costa and Cesc Fabregas earnt the Londoners three Group G points.
The Israelis' goalkeeper Predrag Rajkovic sent Willian tumbling in the box but Hazard blazed the sixth-minute penalty over the bar.
Rajkovic was having a terrible game and on the quarter-hour mark he allowed a Willian in-swinging free-kick, which was missed by everyone, to creep inside his far post.
Costa scored a stunning over the shoulder volley into the top corner from Fabregas's pinpoint cross and the latter than got his name on the scoresheet, tapping home the rebound after Rajkovic saved Loic Remy's shot.
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