England got the victory in Donetsk after co-host Ukraine had a second-half shot that television pictures appeared to show crossed the goal-line before being cleared by defender John Terry. Wayne Rooney, in his return following a two-game ban, scored after 48 minutes for England.
“When the ball may have crossed the line, we got a bit of luck and went on to win,” England manager Roy Hodgson told ITV.
England, which has never won the tournament, will play Italy, while France faces world and defending champion Spain for a place in the semifinals. In the other quarterfinals, the Czech Republic faces Portugal and Germany takes on Greece.
England finished group play with seven points, three more than France, with Ukraine and Sweden on three.
“If someone said we’d win the group I don’t think they’d have many takers,” Hodgson said.
Sweden, already eliminated from the tournament, gave France a first defeat in 24 matches on goals from Zlatan Ibrahimovic and Sebastian Larsson in Kiev.
Ukraine, needing a win to avoid elimination, was denied a goal after 62 minutes that would have tied its game against England.
A shot from Marko Devic was half blocked by England goalkeeper Joe Hart and looped toward goal. Terry got back to hook the ball clear, although replays showed it may have crossed the line before he did so.
Soccer’s world governing body FIFA will decide next month whether to introduce goal-line technology following incidents including a disallowed goal by England’s Frank Lampard against Germany at the 2010 World Cup.
Ukraine was mainly restricted to long shots in the first half, even though it had most of the play, and Andriy Yarmolenko tested Hart with a low 15-yard (13.7 meter) shot after 30 minutes.
England still had the two best opportunities of the half. Rooney headed wide from a cross by Manchester United teammate Ashley Young with only the goalkeeper to beat on 28 minutes and Terry headed over four minutes later.
England took the lead three minutes into the second half. Steven Gerrard’s low cross from the right was deflected by a defender, causing goalkeeper Andriy Pyatov to fail to hold the ball and allowing Rooney to head into an open goal. It was the striker’s 29th goal for his country and first at a tournament since Euro 2004.
Ukraine responded by pinning England back for five minutes, before Gerrard set Rooney free on a counterattack that led to a shot from James Milner.
Artem Milevskiy should have tied the game on 61 minutes when Ukraine took a quick corner and he headed over from six yards.
Moments later the co-hosts came even closer with the effort from Devic.
Pyatov denied Ashley Cole a first international goal on the England fullback’s 97th appearance, while at the other end a swirling shot by Yevhen Konoplyanka was deflected clear by Hart.
“Rooney’s goal was a tap-in but if he keeps scoring tap- ins for us we’ll go a long way,” England captain Gerrard told uefa.com. “He’s important to the team and the nation.”
In Kiev, France had the better of play in the first half, though only put two of its 10 shots on target. The best chances came when France’s Franck Ribery was stopped by goalkeeper Andreas Isaksson after a turnover in the area and Sweden’s Ola Toivonen hit the post after beating defender Philippe Mexes and going around goalkeeper Hugo Lloris.
Sweden took the lead in the 54th minute as Larsson crossed from the right into the center of the area for Ibrahimovic, who twisted and volleyed a shot into the bottom right corner.
Sweden clinched the victory in stoppage time when Christian Wilhelmsson hit the crossbar and the ball rebounded to Larsson, who fired it into the roof of the open net.