Joseph Yobo headed what seemed as if it might be a potentially vital away goal to level the scores on the half-hour mark, but David Moyes' team then collapsed and conceded four second-half goals to slump to their sixth defeat in seven matches this season.
Claudiu Niculescu gave the hosts the lead before Yobo's effort, while second-half strikes from Ianis Zicu and Florentin Petre, together with a Florin Bratu brace, put Dinamo in sight of a place in the group stage of the competition.
Moyes made three changes to the team that lost to Portsmouth last weekend, recalling James McFadden, Leon Osman and the experienced Phil Neville to his starting line-up.
The changes almost paid immediate dividends as Neville played in McFadden after just four minutes, only for the Scotland international to pull his shot wide of the target.
The visitors enjoyed a good spell of pressure in the opening quarter of the match but paid the price for misses by Simon Davies and Yobo when they fell behind in the 27th minute.
A swift counter-attack finished with Petre crossing for Niculescu to bundle the ball home, despite Yobo's attempted challenge.
The Nigerian defender made amends just three minutes later when he rose to head home Davies' free-kick, bringing Everton level and ensuring the scoreline more closely reflected play.
However, the second half proved to be a nightmare for the Toffees. They conceded four, but without some excellent goalkeeping from Nigel Martyn it could have been far worse.
Zicu restored Dinamo's lead in the 55th minute with a well-crafted goal, clipping a pinpoint cross from Niculescu beyond Martyn.
Everton's lone striker Marcus Bent was a virtual spectator as Martyn attempted to keep the score down.
However, a deserved third arrived for the Romanian side after 71 minutes when Petre rifled the ball in from the right.
Four minutes later Bratu smashed a low shot past Martyn to almost certainly end Everton's European hopes for this season.
Moyes introduced Duncan Ferguson in a bid to salvage something from the trip and the switch almost paid off when home goalkeeper Uladzimir Gaev had to claw a powerful header off the line.
There was still time for Bratu to deliver a final blow by grabbing his second and Dinamo's fifth in the last minute.