Rooney, suspended for the first two matches, wasted a good first-half chance but could not miss three minutes into the second when Steven Gerrard's deflected right-wing cross deceived keeper Andriy Pyatov and left the striker an easy header.
England finished with seven points and will play Group C runners-up Italy in Kiev on Sunday. France's 2-0 defeat by Sweden left them second on four points and facing holders Spain.
Ukraine, needing a win to go through, came desperately close to equalising after 62 minutes when Marco Devic's shot was half-saved by Joe Hart then hacked off the line by John Terry.
Ukraine's players and most of the crowd screamed goal but the fifth official decided otherwise and the hosts were left fuming after television replays suggested the ball may have crossed the line.
For the second major tournament England, who were on the wrong side of a similar decision against Germany in the 2010 World Cup, were involved in an incident that will increase the pressure for the introduction of goal-line technology.
The additional assistant referee, introduced to try to deal with close goal-line decisions, did not indicate a goal to the referee, to the fury of Ukraine coach Oleg Blokhin.
Ukraine started without their record scorer Andriy Shevchenko who was on the bench due to a knee injury but that did not stop them dominating the first half.
They had far more possession and laid siege to England's goal for long periods with their first attempt coming after seven minutes when Denys Garmash fired over from long range.
Terry blocked an Evhen Konoplyanka shot that was heading goalwards after 18 minutes and Ukraine skipper Anatoly Tymoshchuk should have done better with the rebound which he also fired over.
A mistake by Ashley Young after 22 minutes allowed Oleh Gusyev to test Hart and the England keeper had to make another smart save eight minutes later, getting down to a shot from Andriy Yarmolenko.
England's first attempt at goal came after 28 minutes when Rooney headed a Young cross wide with only Pyatov to beat.
Rooney made amends after the break though with his simple back post header to put England in front and, having survived the goal-line incident, they had Hart to thank for a fine save to keep out a fierce long-range Konoplyanka strike.