Arsenal scored three second-half goals to take control of their Champions League third qualifying round tie.
Fabregas (right) was in superb goalscoring form for Arsenal
Cesc Fabregas collected a lay-off from Robin van Persie before slotting home from a tight angle after 63 minutes.
Van Persie doubled their lead a minute later after collecting a long pass from Alexander Hleb and Fabregas scored the third after a jinking run.
Jose Antonio Reyes, linked with a move to Real Madrid, was left out of the side for the first-leg encounter.
Arsenal went into their first competitive fixture of the season with a much-changed line-up from the team that started last season's Champions League final to Barcelona.
Thierry Henry, Freddie Ljungberg, Jens Lehmann, Ashley Cole, Robert Pires and Sol Campbell were all missing from the team that lost in Paris.
But summer signing Tomas Rosicky showed some good touches, while the Arsenal team showed great maturity in possession.
Dinamo keeper Ivan Turina saved long-range efforts from Emmanuel Adebayor, Van Persie and Rosicky during an opening half in which both teams were largely restricted to shots from distance.
Adebayor, making his European debut for Arsenal, was heavily involved and went close with a blistering strike that eluded Turina but flew wide of the post.
The home team, playing in front of a capacity crowd, opened up Arsenal on several occasions, with Lukas Modric a real threat.
Modric, who is on a 10-year contract at Zagreb, troubled Arsenal with his direct running and clever touches.
The 20-year-old just missed the target after striking Etto's pull-back with his first touch and later opened up the Gunners defence, collecting a return pass before just shooting wide from a tight angle.
Manuel Almunia saved low to his left to deny Marijan Buljat, while Modric struck the ball wide with a shot across goal under pressure from Johan Djourou.
Dinamo enjoyed a spell of possession after the break and an early ball from the influential Modric released Eduardo, who took his shot early but failed to find the target, while Van Persie forced a scrambling save from Turina with another effort from distance.
But Arsenal's patience in possession was finally rewarded when Van Persie controlled a Hleb pass with his first touch and laid the ball off for Fabregas to score with a precise finish.
Within a minute Arsenal had doubled their advantage, Van Persie collecting Hleb's superb long ball forward before slotting a shot across Turina, who should have got at least a hand to the low strike.
Modric almost pulled a goal back before Fabregas showed great control in dribbling the ball into the Zagreb area before scoring with a shot across goal.
It could have been even worse for Zagreb but Turina saved with his feet from Mathieu Flamini.
Dinamo Zagreb: Turina, Buljat, Cale (Santos Carlos 93), Nowotny, Corluka, Cvitanovic, Etto, Modric, Eduardo, Mamic, Vugrinec (Aglic 27).
Subs not used: Anderson Costa, Drpic, Ljubojevic
Arsenal: Almunia, Eboue, Toure, Hoyte, Djourou, Fabregas, Gilberto, Hleb, Rosicky (Flamini 82), Van Persie, Adebayor (Aliadiere 82).
Subs not used: Poom, Cygan, Traore, Song, Reyes.
Booked: Hoyte, Hleb
Goals: Fabregas 63, 79, Van Persie 64
Referee: Kyros Vassaras (Greece).