Maurizio Sarri breathes a sigh of relief. For now

After the tumultuous week he has endured, you cannot blame Maurizio Sarri for taking just about anything right now. Three days on from his side's meek surrender in the FA Cup fifth-round tie against Manchester United, the embattled Chelsea manager arrived here knowing that anything less than a smooth progression to the last 16 of Europe's secondary club competition was likely to spell the end of his reign.

That scenario, at least, was averted as second-half goals from Olivier Giroud, Ross Barkley and Callum Hudson-Odoi eased Chelsea's passage through the Europa Cup against opponents who never gave up hope that they might rescue something from this tie. But with Pep Guardiola's Manchester City up next in the Carabao Cup final on Sunday – a fortnight on from their 6-0 destruction of Chelsea in the Premier League – and with Tottenham following on Wednesday, Sarri will know there are far greater tests to come before he is safely out of the woods.

Barkley's free-kick to make it 2-0 after the dismissal of Malmö's Rasmus Bengtsson for a second booking raised the biggest cheer from the home supporters on a night when they were consistently outsung by the 3,500 fans who had made the trip from Sweden. Yet even with his side cruising through to the next round, the introduction of Jorginho from the substitute's bench was greeted with more than a smattering of boos – an indication of the level of antipathy that Sarri has faced in recent weeks.

It is certainly not going to be easy to turn those attitudes around under such enormous pressure from above.

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Most of the atmosphere before the game had been generated by Malmö's travelling support in the Shed end, hijacking the traditional pre-match warmup tune The Liquidator with the less than creative ditty of "F*ck you Chelsea, f*ck you". The home supporters did muster a loud cheer for Hudson-Odoi's name when it was announced before kick-off, with the teenager making his first start since a prospective £35m move to Bayern Munich was vetoed by Chelsea in the January transfer window. He had played just 19 minutes in the meantime and, having been left on the bench by Sarri in favour of Davide Zappacosta when his side were desperately chasing the game against United, the youth team product has become the poster boy for the disaffected among Chelsea's support base in recent weeks.

An interview with him in the match programme unsurprisingly made no mention of Bayern's interest but his presence on the pitch – as well as the absence of the under-fire Jorginho and Marcos Alonso from the starting lineup – seemed to have lifted spirits slightly among the home supporters after the misery of Monday night.

That almost evaporated in the 13th minute, however, when Hudson-Odoi's mistake presented Markus Rosenberg with a good opportunity to score. But the striker somehow failed to hit the target. Patrolling the touchline, Sarri took a long, hard suck on the smoked cigarette butt he has taken to carrying during matches for emergencies. At least he comes prepared these days.

The 60-year-old had recognised the frustration of fans at the recent run of poor results in his programme notes and admitted his players needed to show “character and resilience” to get themselves out of their slump. Yet for some, the Italian’s days are already numbered, even if they somehow manage to find a way past City at Wembley.

"It's frustrating," admitted David, a season ticket holder in the Matthew Harding stand. "You can see what he is trying to do but it's just not working. I know he had some success with Napoli but it's very difficult to play the way he wants in this country."

Sarri had also insisted this week that he prefers “a player able to move the ball very fast” as his midfield pivot, meaning that N’Golo Kanté found himself in his less favoured role on the right of the three once again. But after a first half devoid of entertainment that had seen Chelsea’s players leave the pitch to some boos, the Frenchman’s interception and lightning quick break to play in Willian was evidence that he is capable of doing both as Olivier Giroud tapped home the Brazilian’s perfect cross before Barkley and Hudson-Odoi made the result safe. At least Sarri could breathe a sigh of relief. For now.