‘It’s a miracle!” exclaims a Swedish official. No, he is corrected by a beaming colleague: “It’s bureaucracy.” This is a man whose diplomatic pincer skills have just stuck it to the Nazi hate machine and will save tens of thousands of Jewish lives. His name is Gösta Engzell, a real-life bureaucrat in the Swedish foreign ministry during the second world war, played here by Henrik Dorsin as bumbling and avuncular in his comfy cardigans and dicky bow ties.

If we are honest, Engzell’s desk-based heroism – deploying the power of loopholes, paperwork and diplomatic notes verbales – to save lives is not terribly cinematic. Co-directors Thérèse Ahlbeck and Marcus Olsson’s workaround is to give us shots of diplomats dashing along the corridors of power, huffing and puffing; it all adds to the film’s affable comic mood, pleasant enough but sometimes jarring with the seriousness of what is at stake.

Engzell is shown as a man little respected in the ministry; his team is crammed into a tiny basement office, sewage pipes clanking above their heads. Modest and unassuming, Engzell toes the line, processing visas and dealing with immigration issues. The unspoken rule is that applications from Jewish people are archived; their plight is a “non-issue”. Sweden maintained a policy of neutrality throughout the war, but a murkier truth is portrayed here. Engzell’s bosses strain to keep on the good side of the Germans, content to dismiss reports of genocide as “rumours”.

But after the arrival of principled young colleague Rut Vogl (Sissela Benn), Engzell finds his moral compass, and directs his team to process visas for Norwegian Jews. Anyone with a Swedish connection gets a stamp – and it rubs off on others. For good to prosper, it seems, all it takes is enough good people to take action. It’s an uplifting message in a watchable movie.

• The Swedish Connection is in UK and Irish cinemas from 13 February, and on Netflix from 19 February.