Rooney, back after a two-match suspension, headed wide from an Ashley Young cross when unmarked after 28 minutes. Ukraine had more of the possession but England goalkeeper Joe Hart was only tested once, by a low effort from Andriy Yarmolenko.
The Manchester United striker made up for the miss when just after the break he put England in front with a simple header from a badly cleared Steven Gerrard effort.
England’s victory means that they top group D and face Italy in the quarter-finals. France, who suffered a surprise 2-0 defeat at the hands of Sweden, qualify in second place and face world and European champions Spain in the last eight.