Ukraine's win over Poland means that they remain one point behind England, but Hodgson was happy that they gave themselves a little margin for error when they face Poland on Tuesday, although he doesn't plan on using it.
"At least the knife is removed from our throat. Had we drawn or lost, with Ukraine winning, it would have been a major problem," Hodgson told PA.
"If the very worst happens now and we, for some reason, can't reproduce this performance or do badly, we still have a chance of getting to the World Cup. That was important for us.
"We don't count on using that vehicle. We have given ourselves a slight margin for error but we count on beating Poland and getting there under our own steam as top of the group."
Should England win their final group game then they would qualify for next year's World Cup, but anything less than three points is likely to see them finish second as Ukraine face San Marino.