Andros Townsend was among the goals on his debut as the Three Lions recorded a 4-1 triumph at Wembley to move back to the top of Group H, meaning that victory over Poland on Tuesday will guarantee their place at Brazil 2014.
Hodgson has praised the attitude of his players, singling out Tottenham Hotspur's Townsend for special recognition, and has called upon them to reproduce the same time of performance in four days' time.
"It is only a half done job, but it is nice to know whatever happens we will not be out," Hodgson told BBC Sport.
"We want to win, and we believe we can. Our attitude will be exactly the same. It is slightly less of a knife against our throat. There is a big job still to do.
"The players are so desperate to get it done, I have to trust and believe in them that they can perform in the same way. We thought Andros Townsend's pace against a packed defence would open doors. He really performed on the night."
Ukraine are England's closet rivals for qualification, and are one point behind Hodgson's side ahead of the last round of fixtures in midweek, when they face San Marino.