It has been one of the images of the World Cup so far. Eyebrows raised, shoulders back, palms upturned, Vladimir Putin savoured Russia’s breakthrough goal in the opening game against Saudi Arabia before he leant into a handshake with Mohammed bin Salman.

Putin was not finished. When the president’s team made it 5-0 in stoppage time, he held his hands wide apart in the most expansive of shrugs. There was no handshake on this occasion for the Saudi crown prince. With the scoreline as it was, it might have looked patronising.

The first thing to say is that many Russian people absolutely loved Putin’s reactions, which were somehow in keeping with his public image. Big Vlad is the big man over here and there is Putin merchandise across all of the tat stores, plenty of which depicts him looking hard and cool.

The wider point is that, in a heartbeat, Putin caught the mood of pleasant surprise. Russia’s tournament had opened to a backdrop of pessimism, with Stanislav Cherchesov’s team – who had gone seven matches without a win – written off as the worst in living memory. Russia not only won El Gasico, they followed it up by thumping Egypt. There was a return to type in game three, when they were taken apart by Uruguay and, before the last-16 tie with Spain, the fatalism was back. Cherchesov and his players needed a miracle.

After a resolute defensive display, they got one in the shape of the penalty shootout triumph and, in a country that likes to swing from one extreme to the other, the mood has transformed. Now, there is rampant optimism; the feeling that anything is possible. If “Ros-Si-Ya” can beat Spain, then why not Croatia in Saturday’s quarter-final in Sochi? Croatia, certainly, did not look like world-beaters in their last-16 shootout win over Denmark.

Cherchesov was asked whether Croatia was the match of his life, and that of his players. “I hope that the most important games are still ahead of us,” he said, with a shrug of which Putin would have been proud. Cherchesov paused. He had made his point. “Is this a good answer?” he asked. “Brevity is the sister of genius, as Anton Chekhov said.”

The 19th-century playwright could not have scripted Russia’s upsurge and not only because his work tended to be associated with gloominess. Even the greatest writer of fiction would have hesitated to put Russia in a first World Cup quarter-final since 1970, when they competed as the Soviet Union. They are, after all, the lowest-ranked team at the finals. Yet here they are.

Cherchesov said he was “not a daydreamer”. He has a goal and it is to beat Croatia. His watchwords have been focus and balance. “We Russians like extremes but let’s not go to extremes,” Cherchesov said. “The media can be euphoric, not us. Reminding ourselves about Spain is a good story but let’s look forward. Criticism is good. We should have our eyes open and our ears ready and being praised all the time is bad. At the same time, the criticism should be analytical. It should not be to humiliate.”

Cherchesov was joined at the press conference by Mario Fernandes and, if one player symbolises the squad’s capacity to overcome, it is he. The Brazil-born right-back had depression after he signed his first senior contract at Grêmio in 2009, had problems with his diet and also drank and partied too much.

At 21, Fernandes was called up by Brazil but he went on a night out and missed his flight. He was kicked out of the squad. He turned his life around when he moved to CSKA Moscow in 2012 and, although he would be capped by Brazil in 2014, he applied for Russia citizenship on residency grounds and, in 2017, he made his debut for them. Fernandes has excelled at this tournament. “The problems I had are in the past,” he said. “I’m a different man now.”

Cherchesov reported that the attacking midfielder Alan Dzagoev, who has been out with a hamstring injury since the Saudi game, had returned to training and was available. Aleksandr Golovin, however, has impressed as the No 10 in his absence. The left-back Yuri Zhirkov is fighting a calf problem.

What of the other piece of team news? Will Putin attend? His spokesman, Dmitry Pestov, has said that Putin will watch the game in Moscow because he is so busy but it would not be a surprise if he showed up. Putin loves Sochi and he has a residence here. “The president called me before and after the Spain game,” Cherchesov said. “When he supports us, it makes us comfortable. It’s an extra boost in terms of motivation.”