Harry Kane back to his lethal best, productivity from set pieces, yet another clean sheet – the fifth out of five at Euro 2020 – and, for the coup de grace, a first England goal for Jordan Henderson on the occasion of his 62nd cap. This was the night when pretty much everything was picture perfect for Gareth Southgate and his players as they set up a Wembley semi-final against Denmark on Wednesday night.

As England came home from Rome it was easy to wonder whether football was headed in the same direction and, certainly, the fans of the team who had made it from across mainland Europe into the Stadio Olimpico thought that way.

Only five times previously have England got this far at a major tournament and the sense that the squad is peaking at the right time is unmistakable. The pre-match expectations had been sky-high after the epic last-16 win over Germany last Tuesday, and Ukraine, who had scraped out of their group with three points, were viewed as being straight-forward opposition.

It was a potentially fraught combination, the excess before a crash, a scenario we have seen before, but England’s focus was faultless. With an early Kane goal to settle them, they subjected Ukraine to an onslaught in the second half, Harry Maguire and Henderson heading home from set pieces and Kane helping himself to another in between times. The captain has now scored nine goals at major championships – one short of Gary Lineker’s England record.

A note of caution. Denmark are running on emotion and an extraordinary feel-good factor; a nation united after Christian Eriksen’s collapse during their first game against Finland. They have already beaten England 1-0 at Wembley this season and drawn 0-0 with them in Copenhagen.

But what fires English belief is how balanced they have come to look. When a team is this untroubled at the back, and here they permitted Ukraine precious little, the attacking players can have the confidence to know that even one moment from them could be decisive.

There was more than one here as England cut loose. Kane could have had a hat-trick – he caught a left-foot volley with murderous power on 62 minutes, which drew a fine save out of Georgiy Bushchan – and he deserved the man of the match award. But only just.

Luke Shaw was a thrusting presence at left-back from the first whistle and he made Kane’s second, moments after whipping in the free-kick from which Maguire scored his goal. With Raheem Sterling, who was also excellent, drifting inside from the left, Shaw could see clear running lanes up the flank. How he embraced them.

It was possible to count the defensive errors from England on a few fingers. Kyle Walker made two in the first half and got away with them while Jordan Pickford was guilty of a sliced clearance in the 70th minute, having left his area. It led to a shot from Roman Yaremchuk, which Walker blocked. Picking holes? Definitely. England’s assurance was supreme and Pickford was largely untroubled, his save from the substitute, Yevhenii Makarenko, in the 74th minute the only time that he was extended.

Southgate had happy memories of this storied venue, having played in England’s 0-0 draw with Italy in 1997 that secured their qualification to the World Cup the following year. This was way better, with England making the dream start when Sterling sliced inside to thread a beautiful low pass through for Kane, who had spun in behind a static Ukraine backline. He lunged to get to the ball first before lifting it up and past Bushchan.

The spring in Kane’s step was pronounced; the memory of his previous toils at the tournament fading. Southgate had predicted that Kane’s late goal against Germany would act as a catalyst. That feels wise now.

Andriy Shevchenko’s starting 3-5-2 formation came to look more like 5-3-2 and, on 36 minutes, the Ukraine manager made a bold change, replacing the centre-half, Sergiy Kryvtsov, with the winger, Viktor Tsygankov, and going to 4-3-3. It was because England’s control was almost total, save for a lapse on 17 minutes by Walker. He left a back-pass short for John Stones which allowed Yaremchuk to get a shot away from a tight angle. Pickford made a routine stop.

Sterling was in the mood, his pace and twinkle toes causing panic in Ukraine’s ranks. He had several eye-catching bursts before the interval, with one, when he beat Oleksandr Karavaev to cross, leading to a Declan Rice blast that worked Bushchan.

Ukraine flickered briefly in their new system, with Walker allowing Yaremchuk to run and cross, Stones once again tidying up, but England should have been further ahead on 40 minutes when Jadon Sancho, starting for the first time at the finals, shot straight at Bushchan following a Shaw cross. Shaw had been flagged offside, although replays showed that he was on.

England powered to their biggest ever knockout phase victory with a devastating burst at the start of the second half, Maguire bullying the hapless Mykola Matvienko to score and Shaw teeing up Kane after a Mason Mount run and Sterling’s impish flick.

Southgate could afford to withdraw the outstanding Rice and then Kalvin Phillips, who were carrying yellow cards – they are now wiped – and he could enjoy a second set piece conversion of the night and the tournament when Henderson guided home from Mount’s corner. England march on.