Rooney header sends England through
Rooney headed home a Steven Gerrard cross when he was left unmarked at the far post after 48 minutes to celebrate his return to the side after serving a two-match ban.
That was enough to give England first place in Group D and secure a clash with Italy in Kiev on Sunday.
Ukraine, who dominated possession for most of the match, thought they had levelled just past the hour mark when Marco Devic's shot was initially parried by goalkeeper Joe Hart.
The ball looped back towards goal before it was hacked away by defender John Terry, although television replays suggested the ball may have crossed the line.
If it did, England had a huge helping hand from the soccer gods who looked the other way in Bloemfontein two years ago when Frank Lampard scored a 'goal' against Germany in the World Cup.
That effort in South Africa definitely did go over the line although it was inexplicably missed by the referee and linesmen.
England coach Roy Hodgson said Devic's shot on Tuesday had probably crossed the line but told reporters it was about time England's luck turned after they also had a 'goal' harshly disallowed against hosts Portugal in Euro 2004.
"We've suffered with bad luck in the past, against Portugal and Germany, maybe it was a case of if there was going to be luck, we were going to get it."
As if to emphasise that perhaps England's fortunes are turning, they held on to win against the host nation in a major tournament for the first time in six attempts since they beat Switzerland in the 1954 World Cup finals.
Ukraine had to beat England to stay in the tournament but, despite dominating possession for most of the first half, the co-hosts allowed their opponents back into the game after the break just as they had done against France on Friday.
They did go close to scoring in the second half though, first when Artem Milevskiy somehow missed a header from point-blank range and then when Devic's shot was cleared off the line.
Despite the introduction of talismanic striker Andriy Shevchenko, who has called time on his international career, Ukraine could not score and joined co-hosts Poland on the sidelines with almost two weeks of the tournament left.
What Ukraine lacked was a player with the driving force and determination of England skipper Steven Gerrard who had at least as big a part in the win as Rooney, who took his England tally to 29 goals on his 75th appearance for his country.
Gerrard, earning his 95th cap, set up Rooney's goal with a cross that via two deflections bamboozled Ukraine's defenders and goalkeeper Andriy Pyatov. England have scored five goals in the tournament so far and Gerrard has created three of them.
He is also playing with the kind of freedom and confidence that has been a feature of his game throughout his long career.
With the debate about whether Gerrard can play in the same midfield as the absent and injured Frank Lampard seemingly over, he defended well, crafted openings and inspired England.
Hodgson's side are now playing more like a club team than the band of disparates England have often seemed in the past.
Coming into this tournament unfancied and with expectations at an all time low because the manager has only been in charge for six weeks, the weight is clearly off their backs.
They may not win this tournament but after their comeback win over Sweden and victory against Ukraine, no-one should bet against them.
Tuesday's hosts have still never won in Donetsk after seven attempts - five at the Donbass Arena - and ended up second-bottom of the group. But they have little time to rebuild.
Ukraine meet England again in a World