Vincent Kompany, so luckless with injuries, has been missing for most of Pep Guardiola’s reign at Manchester City but the club’s captain strode to the fore when it mattered. The Belgian broke Southampton’s resistance by powering a header into the net 10 minutes into the second half, setting the stage for a City win that was embellished by late goals from Leroy Sané and Sergio Agüero.

The victory lifted City to third in the Premier League and widened the buffer zone between them and Manchester United, who are fifth but are, according to Guardiola, capable of gatecrashing the top four.

The Catalan said that Chelsea and Tottenham Hotspur are uncatchable but that any two out of City, United, Liverpool and Arsenal could claim the other qualification spots for next season’s Champions League. After the weekend’s FA Cup semi-final, City host the Manchester derby, which their manager billed as “the real final”.

“Until [City’s] last game at Watford the qualification for the Champions League will be so tough,” said Guardiola. “Two teams are already done and there are four top teams for two positions – Tottenham is so strong, they are not going to drop points. Thursday week we play the real final against United. That’s why it was so important to win [at Southampton] and for our confidence for the semi-final of the FA Cup.”

Failing to challenge for the league title or the Champions League might be grudgingly accepted in Guardiola’s first campaign but dropping out of the top four would be much harder to bear even in a season of transition – especially if City were to be replaced by their neighbours in red. Losing points at St Mary’s would have been infuriating for Guardiola. Slack finishing briefly made it seem likely.

Agüero frittered away two chances in the first five minutes. The first came courtesy of a lovely cross from the left by Gaël Clichy. Agüero showed dexterity to meet it, raising his right leg high like a ballerina before shooting like one, too. Moments later he was put clean through again but fired wide. It took the hosts 10 minutes to retort and then they, also, had cause to lament scruffy finishing. The move had been admirably neat, Nathan Redmond piercing City’s defence with a pass to Manolo Gabbiadini, who calmly cut it back to Dusan Tadic. The Serb’s shot landed in the upper reaches of the Northam Stand.

That Tadic had found such space in front of goal raised concerns about City’s defence, from which John Stones was absent due to an unspecified injury that Guardiola suggested was not too serious. The concerns about City’s defence turned out not to be serious here either, as Kompany and Nicolás Otamendi restricted the hosts to few openings after Tadic’s miss.

City still struggled to score, however. David Silva shooting into the side-netting from close range after a dinked cross by Agüero.

In the second half City raised the tempo. Fraser Forster had to excel to tip over a ferocious long-range shot by Jesús Navas. Kompany ensured that merely delayed the breakthrough. From the resultant corner he outjumped Maya Yoshida and butted a header into the net from six yards.

Sané and Agüero then killed the contest by finishing a pair of deadly counterattacks, each one orchestrated by Kevin De Bruyne.

The breakthrough goal was Kompany’s first since August 2015 and came on only his fifth league appearance of another injury-afflicted campaign. He celebrated his goal like a man giving vent to more than a year’s worth of pent-up fury but otherwise he channelled his anger into a polished performance. That came as no surprise to Guardiola.

“Manuel [Pellegrini] last season missed him a lot last season and we’ve missed him a lot [this season],” said Guardiola. “He’s a guy with the quality to win duels one against one ... And he helps us in defensive and offensive set pieces because he’s a real defender.”

Asked whether Kompany will remain important in the future Guardiola replied: “There is no doubt about it.” But he then added that, of course, plenty of doubt remains. “The problem is when a guy is injured for two or three years, what can we do?

“In the past two years he has had a lot of problems and of course this is only his third or fourth game of the season and that’s not too much – but we are so happy he has come back and hopefully we can take care and he will be fit until the end of the season.”