These are difficult days for Moyes, who led the club to fourth in the Premiership table beyond everyone's expectations last season.
Everton are finding it difficult to win a match now - the 5-1 defeat last night was their sixth in seven games. During this spell they have exited the Champions' League qualifiers - now they are on the point of being out of Europe before the end of the month.
Last night Moyes refused to condemn his players publicly but admitted facing an uphill task in the second leg on 29 September. "There will be no public condemnation of any of my players - but privately we will speak together and we will stick together," the manager said.
"In the first half we did a good job and we could have been in front - but we were not at it in the second half. I feel for the fans and it's my responsibility to raise morale in the camp - it's my team out there." Moyes added: "I'll have to see what I can do to make things better, but there are other important games to think about [at Arsenal on Monday]."
David Weir, the Everton captain, commented: "We're sorry for the fans. But it's not acceptable and we are very disappointed."
Everton fell apart and conceded four goals in the second half. Joseph Yobo had almost put Everton ahead after 18 minutes when he skipped passed Ianis Zicu and rattled in a shot that Uladzimir Gaev did well to turn around the post.
But Everton were stunned in the 27th minute by a swift counter-attack. Florentin Petre delivered the cross and Claudiu Niculescu bundled the ball home despite being under pressure from Weir.
Everton's response was emphatic - they drew level three minutes later with what at the time looked to be a priceless away goal. Yobo got on the end of an inswinging free-kick from Simon Davies to head powerfully beyond the goalkeeper Gaev to make it 1-1. It was the Nigerian defender's first goal in 16 months and he savoured the moment as did the Everton fans sprinkled around this old-fashioned arena.
But the Romanians regained the lead in the 52nd minute. Niculescu delivered a superb cross into the path of Zicu, who got a marvellous first touch to clip a shot beyond Martyn.
Dinamo were beginning to dictate the pace and Zicu was growing in confidence. He latched on to a through ball from Stefan Grigorie but Martyn raced off his line to smother the ball.
This was a worrying spell for Everton. Dinamo's second goal had rattled the visitors, they were showing little of their early fluency and were having difficulty providing Marcus Bent with the kind of service that a lone striker needs. They had lost the midfield battle with James McFadden in particular, making his first start of the season, unable to get any quality crosses in.
Dinamo added a third in the 71st minute with a stunning strike by Petre. This was becoming a rout and an embarrassment for Everton as they began to buckle under a wave of attacks. Their problems increased in the 75th minute when Dinamo extended their lead to 4-1.
Again it was a spectacular goal, this time from the substitute Florin Bratu as he delivered an unstoppable shot beyond Martyn. Duncan Ferguson almost pulled one back, only for Bratu to get a fifth for Dinamo right on full-time.
Dinamo Bucharest (4-4-2): Gaev; Goian, Tamas, Moti, Christian Pulhac; Petre (Galamaz, 83), Margaritescu, Plesan (Munteanu, 79), Grigorie; Zicu, Niculescu (Bratu, 61). Substitutes not used: Guso, Balan, Cristea, Vlad Baltoi.
Everton (4-5-1): Martyn; Hibbert, Yobo, Weir, Nuno Valente; Osman, Neville, Cahill, Davies (Kilbane, 73), McFadden (Ferguson, 78); Bent. Substitutes not used: Wright (gk), Ferrari, Li Tie, Arteta, Hughes.
Referee: G Paparesta (Italy).
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