There had been worries about Wayne Rooney's boiling point, but after a day of scorching heat in Donetsk it was his coolness that delivered England the result which enables them to avoid the defending champions in the quarter-finals.

The ambient temperature had dropped significantly by the time Tuesday night's match kicked off, but for England it felt like an unbroken heatwave as Ukraine attacked at pace, in numbers and without respite. Without luck, too, as it turned out. The co-hosts, whose principal ambition was to advance beyond the tournament's group stage, transformed the stadium into an oven in which England's hopes were in danger of evaporating, but Rooney's goal, three minutes into the second half, rewarded his team-mates for their refusal to yield.

Every time the home team drove towards the English penalty area, a noise rose from the grandstands like a wall constructed in one of the city's steel plants. The white-shirted defenders could not hear themselves think, never mind communicate with each other. But if John Terry was the figure around whom the defence massed, and whose interventions kept them in the game, Rooney added authority to the attack.

Despite the incessant pressure, he should have given England the lead even earlier, just before the half-hour, when the Ukrainian onslaught was at its height. Terry supplied Ashley Young with an excellent ball and the cross from the left found Rooney at the far post. But, having freed himself from the attentions of Yaroslav Rakitskiy and found clear space for his header, he could not hit an open target.

There had been a fear that, restricted to a mere 37 minutes of international football in the past eight months, Rooney might either be lacking sharpness or react to his much heralded return with the sort of indiscriminate release of explosive energy that has damaged England in the past, most recently last October, when his kick at a Montenegrin opponent cost him his place in last week's opening matches against France and Sweden.

Better that it should be the former, and there was a definite ring-rustiness to that first header. Yet although England's forwards were on the shortest of rations in the early stages, there was still the occasional subtle linking touch to Young or Danny Welbeck from the returning No10, on whose shoulders such a weight of responsibility had been heaped, not least by the words of his own manager.

Roy Hodgson seems to have decided that the most effective way to get Rooney playing at his best as quickly as possible is to tell him, via repeated public pronouncements, that the team's fortunes depend on him. It is an interesting approach, and the manager must have given it a great deal of thought. During the buildup to the match he even likened Rooney's potential contribution to the way Pelé inspired Brazil to their tournament victories. "Let's hope that Wayne Rooney can start to do that for us on Tuesday night," he said. "Then, if we win, who knows? If Wayne can produce his best, then he can help us keep going even further."

Sven-Goran Eriksson was the first England manager to make the comparison, noting the way the 18-year-old Rooney's impact on the Euro 2004 finals resembled that of the Brazilian on the 1958 World Cup at the same age. Rooney, however, is still waiting to transfer that promise into achievement; last night he was winning his 75th cap and, at 26-years-old, was hoping that his fourth appearance in a major tournament would finally be the one to justify the reputation.

For Hodgson, Rooney's best work is done as the second striker, and when the manager was deciding who would be his most effective partner against Ukraine, the statistic that interested him the most was the one that said Rooney had scored 17 goals in the 20 Premier League matches he and Danny Welbeck started together for Manchester United last season.

With Young, a third player from Old Trafford, on the left wing, England always had a better than usual chance of producing joined-up attacking. If there were few opportunities during that torrid opening half-hour, there were certainly signs of promise. And Hodgson would have noted his superstar's alertness when Steven Gerrard regained possession from a half-cleared corner in the 48th minute and curled in a diagonal cross that took two deflections before bouncing up in front of Rooney, who was waiting on the far post and nodded the ball into the net from a few inches out.

"All credit to Wayne," Gerrard said afterwards. "There was a lot of pressure on him tonight, and he took it. Sometimes a tap-in is just as important as a 30-yarder." Once again, too, a Gerrard cross from the right had set up an England goal: the third example in this tournament, following those scored by Joleon Lescott against France and Andy Carroll against Sweden.

Not surprisingly, Rooney did not have all his weaponry on display. He sprinted to the touchline before the kick-off, getting his legs warmed up, but five minutes after giving England the lead he was caught and dispossessed by Andriy Yarmolenko, having sprinted clear to meet another ball from Gerrard. He may still not be close to his full capacity by the time England's interest in the tournament comes to an end. But he did enough last night to suggest that Hodgson was not wasting his breath.