The Ukrainians thought they had equalized in the 62nd minute when Marko Devic's looping shot appeared to cross the line before it was hooked clear by John Terry. But the goal was not awarded by the referee or his extra assistant behind the goal.
The controversy will likely add to the pressure on UEFA President Michel Platini to reverse his opposition to goal-line technology.
"I was sad because the ball was in the goal by one meter," Ukraine coach Oleg Blokhin said.
Rooney, who sat out his team's first two matches at Euro 2012, returned from suspension Tuesday and headed in the lone goal in the 48th. Steven Gerrard whipped a cross into the penalty area that went past two defenders before Pyatov let it slip through his hands as he crouched to collect the ball. Rooney was primed at the far post and put England ahead.
"There was a lot of pressure on Wayne," Gerrard said. "But he stood up and when we needed him he was there."
It was Rooney's first goal at a major tournament since scoring four at Euro 2004 as an 18-year-old rising star. He failed to find the net at both the 2006 and 2010 World Cups, while England failed to even qualify for Euro 2008.
With the victory, England won Group D and will next face Italy in Kiev on Sunday. France also advanced, despite losing to Sweden 2-0 in the other group match. Ukraine, however, will have to watch the rest of the tournament as spectators, just like co-host Poland.
"No one believed in us at the start but we're gaining momentum at the right time," Gerrard said. "I don't think tonight's performance was fantastic, but we stuck together, got the job done and won 1-0 against a good team.
"We've been criticized in the past maybe for not turning up, underperforming, and we can take that. We're man enough. But you saw a reaction. When you get criticized it's not nice and you have to react by playing better and I think that's what we did."
England is now unbeaten in five matches under Roy Hodgson, who took over the team last month following Fabio Capello's resignation in February.
"Most people didn't think we'd get out of the group," Hodgson said. "We've won it on points instead of goal difference as well. This was an away game with a capital 'A.'
"There were (50,000) people in and only 4,000 brave England fans. When we needed a bit of luck, when the ball may have crossed the line, we got a bit of luck and went on to win the match."
A draw wouldn't have helped Ukraine advance, but it would have given the team hope knowing that a victory was enough to put the co-hosts through to the quarterfinals.
Ukraine had never won at the Donbass Arena on four previous occasions, and failed to seize on its dominance when it mattered most.
With their fans dominating the crowd and captain Andriy Shevchenko on the bench at the start due to a left knee injury, the Ukrainians were in control from the outset, far more confident on the ball and containing England inside its own half.
But early efforts were from distance, with Denys Garmash and Yaroslav Rakitskiy striking over, while England midfielder Scott Parker had to throw his body into the path of Devic's low shot.
They were getting closer, though, and looking more even menacing going forward with Oleh Gusev getting into the penalty area and sending in a shot that just creep over Joe Hart's goal.
Before Rooney scored, England came close to testing Pyatov when Ashley Young sent in a low cross that was easily dealt with in the 11th minute.
As Ukraine attacked the England goal, Rooney was quiet in the first half of his first competitive match in more than a month until he sent a free header wide in the 28th from Young's cross.
At the 2010 World Cup, England midfielder Frank Lampard was denied a goal that would have leveled the match against Germany at 2-2. England ended up losing 4-1.
"To be successful you need a bit of luck," Gerrard said. "Two years ago we were unlucky with Frank Lampard and we had to pack our bags."
