Rooney, suspended for the first two matches, scored three minutes into the second half when captain Steven Gerrard's deflected right-wing cross deceived goalkeeper Andriy Pyatov and left the returning striker with an easy header.
England finished as Group D winners with seven points and will play Italy in Kiev on Sunday. France's 2-0 defeat by Sweden left them second on four points and facing holders Spain.
However, England's victory only came after Ukraine, needing a win to go through, were desperately close to levelling the scores just past the hour mark when Marco Devic's shot was half-saved by Joe Hart then hacked off the line by John Terry.
Image: England's Scott Parker celebrates with teammates at the end of the Euro 2012 soccer championship Group D match between England and Ukraine in Donetsk, Ukraine, Tuesday, June 19, 2012. England won 1-0.