But Olomouc once again breached the Aberdeen backline early in the second leg encounter.
Chris Maguire twice threatened a quick reply for the visitors before Olomouc doubled their lead on the night.
But Olomouc would be made to regret their profligacy and Everton soon took control of proceedings.
Lithuania, with whom Scotland drew 0-0 in Kaunas on Friday, caused a surprise by defeating Czech Republic 1-0 in Olomouc.
Arriving shortly before the break, the goal ended any hopes Olomouc harboured of a unlikely comeback and they were barely given a sniff in the second half.
As in the first leg, when they were forced to weather a spell of sustained Olomouc pressure, the Toffees looked some way off the pace during the opening exchanges.
Everton shrugged off a 6-1 thrashing by Arsenal at the weekend and the furor surrounding defender Joleon Lescott, who was rested, to beat Sigma Olomouc 4-0 at Goodison Park.
While much of the enthusiasm that followed Mark McGhee's appointment evaporated during the humbling by Sigma Olomouc in the Europa League qualifier, there is no need for Aberdeen to panic.
Moyes would have been pleased with his team's overall performance against a talented Olomouc side, especially considering that from the sixth minute onwards they played with 10 men in scorching temperatures.
He played 90 minutes in the Toffees' 6-1 home defeat by Arsenal on the opening day of the Premier League season but was then dropped for their Europa League match against Sigma Olomouc.
Everton drew 1-1 at Czech side Sigma Olomouc for a 5-1 aggregate victory despite playing all but the first seven minutes of the match with only 10 men after full-back Tony Hibbert was sent off for a professional foul.
应用推荐