Wayne Rooney ’s headed goal early in the second half put England top with seven points and they will face Group C runners-up Italy in Kiev Sunday.
France, who ended with four points, one more than Ukraine, will face holders Spain in Donetsk Saturday after surrendering top spot when Zlatan Ibrahimovic and Sebastian Larsson gave the already-eliminated Swedes a victorious finale with superb goals.
Wednesday is a rest day at the tournament with the quarterfinals beginning Thursday when the Czech Republic face Portugal in Warsaw.
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.
Ukraine, needing a win to go through, came desperately close to equalizing 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.
England’s first attempt at goal came after 28 minutes when Rooney headed a Young cross wide with only Pyatov to beat.
Below-par France qualified for the quarter-finals as Group D runners-up despite a superb Zlatan Ibrahimovic strike and a late Sebastian Larsson goal condemning them to a 2-0 defeat by Sweden.
Ibrahimovic’s acrobatic volley from just inside the area, the striker swivelling to meet Larsson’s cross, put already-eliminated Sweden ahead on 54 minutes in what will be a strong contender for goal of the tournament.
It was Sweden’s first victory over the French since 1969 and France’s first defeat in 24 games.
“Sweden beat an average France team,” France coach Laurent Blanc told French television. “They were better in all departments.”
Already eliminated, the Swedes had clearly decided they were not going down without a fight and they flew into challenges from the first whistle.
With the French through as long as Ukraine did not win, there was always the chance of a dull game but both sides played with commitment, pace and skill.
France looked slick in possession and alarm bells rang every time Franck Ribery ran at the Swedish defense. Several times he ghosted down the left wing to create openings that were wasted by his teammates.
Ribery, Hatem Ben Arfa, Karim Benzema and Samir Nasri all threatened with shots from distance but none troubled Andreas Isaksson unduly.
Ibrahimovic brought the crowd to their feet in the 54th minute, finishing Larsson’s cross with a breathtaking scissor-kick.