The Merseysiders battled hard to overturn a 5-1 deficit from the away leg of this first round tie, but a solitary Tim Cahill goal was the only reward their efforts.
David Moyes made three changes to his side following their shoddy performance in the 1-0 defeat by Wigan, with Duncan Ferguson, Matteo Ferrari and Tony Hibbert all recalled to the starting line-up.
Surprisingly captain David Weir was one of the men dropped, with Phil Neville taking the armband.
All-out attack was Everton's tactic from the off and it made for an entertaining opening spell, as they also left gaps at the back for the visitors to exploit.
Gabriel Tamas tested Nigel Martyn with a free-kick and Stefan Grigorie fired narrowly wide with a long-range effort, but for the most part it was the home side going forward.
James McFadden was a constant threat to the visiting defence, but both he and Ferguson wasted good chances to establish an early lead.
The Toffees finally made the breakthrough in the 29th minute when Ferguson headed Mikel Arteta's cross back across goal for Cahill to nod home.
There was no let-up from Moyes' side and Cahill thought he had doubled the advantage shortly before the interval when he put the ball in the back of the net, only for play to be called back for a foul on Dinamo goalkeeper Uladzimir Hayeu by Ferguson.
Everton piled on the pressure in the early stages of the second half but struggled to create clear scoring opportunities. Nuno Valente and Arteta put plenty of balls into the box but despite causing a measure of panic in the visiting rearguard, they ultimately proved ineffective.
Having seemingly decided they would be unable to withstand the pressure if they just sat back and defended, Dinamo launched a series of attacks.
Florentin Petre had two good chances to equalise, but saw a shot blocked by Joseph Yobo before Martyn tipped a drive over the crossbar.
With 22 minutes to play Moyes introduced strike duo Marcus Bent and James Beattie from the bench, but the move had little impact on the game.
Beattie had one good chance, but Hayeu comfortably dealt with his deflected shot.
Despite having four forwards on the field, Everton were unable to breach the visiting defence for a second time and a disappointing night was made even worse when Arteta was stretchered off in the closing minutes following a bad foul by Mihaita Plesan.