Hat-trick of headers
from everyone really, the perfect night for us," Kane added.
Their display at the Stadio Olimpico was a step-up in class in the final third to previous games at the Euro and they will be favourites at home against a Danish side who played their own quarterfinal against the Czech Republic on Saturday in distant Baku.
This will be the only match England play away from home in the competition and, with Italy currently imposing a five-day quarantine on arrivals from the United Kingdom, the number of England fans in Rome was limited.
However, they still made themselves heard in the crowd of under 12,000.
They had plenty to celebrate, unlike their Ukrainian counterparts, as Andriy Shevchenko's team came up short in their bid to take the country to a first major tournament semifinal.
"We can be proud. Obviously we didn't get the result we wanted but we are on the right track," insisted Shevchenko, who said England had been 'pragmatic'.
Ukraine's chances of shocking England looked dead and buried when they fell behind early on, as Sterling played in Kane who poked the ball past Georgiy Bushchan.
Ukraine's giant striker Roman Yaremchuk forced a save from Jordan Pickford and a Declan Rice piledriver was kept out by Bushchan, with England looking comfortable.
However Ukraine were a different proposition after injured defender Serhiy Kryvtsov was replaced by Dynamo Kiev winger Viktor Tsygankov in the 36th minute.
They finished the first half strongly. England scored again less than a minute after the restart when Luke Shaw delivered a free-kick from the left for Maguire to head in.
Four minutes after that Sterling supplied the overlapping Shaw and he crossed for a rejuvenated Kane to head home. The Tottenham star nearly had his hat-trick, a stinging volley producing a fine save from Bushchan.
From Mason Mount's resulting corner came the fourth goal, another header, this time from Henderson, the first of five substitutes sent on by Southgate who would have been thinking about the semifinal long before this quarterfinal was officially over.
"To come here, prepare differently, play in a historic stadium on a night that was beautiful for football with the weather inspired the players," said Southgate.
"They gave a fantastic performance, not only the players who started but the players who came into the game."
"The recognition in moments like tonight was that we didn't want to take a backwards step. We really wanted to grasp an opportunity. I thought the players were decisive and ruthless all night," he added.