The Edinburgh outfit were completely outclassed by the Croatian champions, who all but finished their Europa League play-off tie inside an hour of the first leg at the Maksimir Stadium.
Goals from Mario Mandzukic, Dimitrios Papadopoulos, Ivica Vrdoljak and former Liverpool midfielder Igor Biscan put Dinamo within touching distance of the group stage ahead of next week's return match.
It could have been even worse for the visitors, who looked clueless throughout as their slick opponents created and missed chance after chance.
Hearts recalled the fit-again Lee Wallace at left-back, while centre-back Marius Zaliukas started in the midst of his four-match domestic ban.
Christian Nade was a lone front-runner in a 4-5-1 formation as the visitors attempted to keep it tight in midfield.
As feared, the pitch was in terrible condition, having been relaid 48 hours earlier following a U2 concert.
Two Dinamo players soon lost their footing on the sandy surface but Pedro Morales showed great balance to help lay on the fifth-minute opener, nutmegging Ruben Palazuelos before feeding Jorge Sammir on the right touchline.
The winger's deep cross was completely misjudged by Marian Kello, allowing Mandzukic to prod home at the far post.
Hearts were struggling to get out of their own half but caught a break when Oeliton Etto left the field on a stretcher in the 18th minute to be replaced by Ivan Tomecak.
Yet, it should have been 2-0 twice in a matter of seconds.
First the completely unmarked Papadopoulos' tame header from Santos Carlos' left-wing cross was pushed behind by Kello and Mandzukic then nodded straight at the keeper from a Morales corner, Hearts once again failing to mark in the box.
Morales was running the show, the Chilean unleashing a wicked dipping shot which clipped the top of Kello's crossbar.
A second goal was looking inevitable and it arrived in the 36th minute courtesy of a comedy of errors.
Ismael Bouzid nodded a Mandzukic knockdown straight to Vrdoljak, whose follow-up was parried by Kello to Papadopoulos for the simplest of finishes.
Morales was then a whisker wide with a tremendous 30-yard free-kick before Hearts' unhappy half ended with Nade being cautioned for a lunging tackle on Vrdoljak.
Manager Csaba Laszlo tried to wrest the initiative by throwing on Ian Black for Bouzid at the restart.
But Dinamo showed no sign of letting up, Morales twice trying his luck from just outside the box.
With 50 minutes gone, Hearts finally had an attempt at goal, Suso Santana's near-post strike pushed behind by Tomislav Butina.
Mandzukic almost scored a picture-book goal when he bamboozled Jose Goncalves and Zaliukas in the box before slipping and scuffing his finish wide.
Moments later, in the 56th minute, Vrdoljak delivered the coup de grace, curling a beautiful finish beyond Kello after a swift counter.
Nade was withdrawn for Gary Glen before Eggert Jonsson deflected Mandzukic's goalbound shot over.
But the respite was brief as the unmarked Biscan headed in the resulting corner, the tie looking over before the hour mark.
Andrej Kramaric replaced Sammir for Dinamo before Zaliukas talked himself into a booking.
Kello's sprawling one-on-one save and a clearance off the line from Jonsson prevented Dinamo making it 5-0 with a quarter of the game remaining.
The outstanding Morales was withdrawn 15 minutes from time, while Papadopoulos hit the post from Tomecak's cross late on.