Job done for Mikel Arteta. After the week he has had, the last thing the Arsenal manager would have wanted was another frantic finale to a home fixture. For much of this contest, after Declan Rice had fired the hosts in front after only 72 seconds, it seemed the Gunners may have to go the distance as they huffed and puffed but couldn’t find a second goal to see off a spirited Dinamo Zagreb side managed by their defensive master Fabio Cannavaro.

A collective sigh of relief greeted the second goal from Kai Havertz – his 14th of the season – when it came 20 minutes before the end, as Martin Ødegaard added a third late on to virtually ensure Arsenal will finish in the top eight and avoid the need for a playoff next month. With a warm‑weather training camp in Dubai lined up instead, after their early exit from the FA Cup, it could do Arteta’s weary squad the world of good in the long run.

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“It’s a massive step – this puts us in a really strong position ahead of our final match against Girona next week,” Arteta said. “If we earn the right to do [the training camp] that would be great. We’ve had a busy schedule and there have been a lot of demands on the players so if we can achieve that it would be great.”

While their progress in this competition has been pretty smooth so far, there was a reminder of fans’ frustrations with Arsenal’s mixed fortunes in the Premier League when it was revealed that a message had been added to a mural outside the ground featuring the co-chair Josh Kroenke and the former midfielder Santi Cazorla. “Time to splash some cash Josh,” it read. Kroenke was watching from the stands as Oleksandr Zinchenko performed well on only his fourth start of the season despite rumours he could leave the club this month, but saw Raheem Sterling struggle until he was replaced by Ethan Nwaneri in the second half.

The England forward was signed on loan from Chelsea at the last minute during the summer but has contributed only one goal so far, with Arsenal now expected to attempt to bolster their squad having been linked this week with a move for Wolves’ Matheus Cunha.

Dinamo have recovered well from the 9-2 mauling they endured against Bayern Munich in their opening match and find themselves level on points with Manchester City with still a realistic chance of a playoff, with one game to play against Milan. But having not played competitively since mid-December because of the winter break in their domestic league and under the command of the former Italy defender Cannavaro for the first time since he replaced Nenad Bjelica, it was perhaps no surprise they took time to find their feet. Arteta had urged his players to take out the frustrations of their draw here against Aston Villa on Saturday and they took him at his word.

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Little more than a minute had passed when they found a way through Dinamo’s defence after Gabriel Martinelli’s cross was teed up expertly inside the area by Havertz for Rice to lash into the net. Rice and Gabriel Magalhães could not take further opportunities, with the Brazilian defender somehow missing the target from a corner routine that led to a mass of mocking arms being waved by the boisterous travelling support behind the goal.

Cannavaro described this beforehand as the “biggest match of my coaching career” and was pleased his side made it to half-time without further damage.

Sterling went into the book at the start of the second half after struggling to spring on to a through-ball and instead clattered into Maxime Bernauer. He was joined by Jurrien Timber soon after as Dinamo succeeded in disrupting Arsenal’s rhythm. Cannavaro said: “I asked them to work together and the second half was much better. But playing against this level of team is not easy.”

Arteta turned to Nwaneri to provide some inspiration off the bench and the 17-year-old was given a rousing reception when he replaced Sterling, having missed the past three weeks with a muscle injury. But Martinelli was the architect of their second goal when he cut in from the left flank and picked out Havertz for a simple header that helped to settle his manager’s nerves.

It was the 500th goal Arsenal have scored since Arteta took over as manager five years ago. Ødegaard made it 501 in stoppage time as he swept home a cross from Leandro Trossard to all but confirm their progress to the knockout stages for the second season in succession, after losing in the quarter-finals against Bayern Munich last year.