Saha (right) was in clinical form in front of goal
Jack Rodwell and Louis Saha scored two goals apiece to put Everton in a strong position after their Europa League first leg tie with Sigma Olomouc.
Tim Howard was lucky not to concede a penalty when he brought down Jakub Petr before Saha volleyed in Marouane Fellaini's cross to open the scoring.
Jack Rodwell then lashed in from 20 yards before marking his European debut with a superb drive for his second.
Saha sealed a fine win with a low, angled shot from just outside the box.
The result virtually takes Everton past their Czech opponents and into the Europa League group stages.
But it was a victory that looked in doubt in the early stages as Everton started in the kind of form that had seen them hammered 6-1 by Arsenal at the weekend.
And had Olomouc been awarded any of three solid shouts for a penalty, the complexion of the tie might have been very different.
There was a change in personnel from the weekend - centre-back Joleon Lescott watching his team-mates from the stands after being publicly criticised by manager David Moyes - but their defence looked just as porous early on.
Olomouc sensed their opponents' hesitancy and were quickly into their stride, making inroads with some slick passing moves.
One such attack could have brought the Czech side a penalty when it looked as though Howard had upended Petr.
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