Louis Saha also weighed in with a brace, as manager David Moyes got the response he was looking for after the 6-1 drubbing at the hands of Arsenal last Sunday.
Moyes recalled both scorers, dropping Joleon Lescott for a poor attitude, while skipper Phil Neville was drafted into defence with Saha replacing Jo against the Czechs, who had defeated Aberdeen 8-1 on aggregate in the previous round.
After a nervous start when goalkeeper Tim Howard parried a shot from Michal Ordos, Everton flexed their muscles and the return leg next week will surely be a formality.
The Toffees almost took the lead in the 23rd minute when Leon Osman rattled in a shot only for Petr Drobisz to turn it over the bar.
Everton's pressure paid off in the 34th minute when Saha got ahead of the goalkeeper to turn Marouane Fellaini's cross into the corner of the net.
The home side added a second six minutes later with a thunderous 20 yard effort from Rodwell after a free-kick by Leighton Baines had come back off the wall.
Moyes' went 3-0 in the 54th minute thanks to another thunderbolt from Rodwell. Baines' free-kick was only partially cleared and Steven Pienaar slid it back into the path of the teenager, who showed little hesitation to rifle a shot from 25 yards beyond a despairing Drobisz.
It was becoming a stroll for Everton with the visitors tiring and Saha added a fourth in the 74th minute with a sweet finish after he was set up by Osman.
Three minutes later Rodwell was given a standing ovation when he was replaced by another bright youngster Dan Gosling, as Everton surely secured their passage into the group stages.