Ms Andersson was elected as prime minister earlier on Wednesday because under Swedish law, she only needed a majority of MPs not to vote against her.
A hundred years after Swedish women were given the vote, the 54-year-old Social Democrat leader was given a standing ovation by sections of the parliament, or Riksdag.
Her election at the head of a minority government followed an 11th-hour deal with the opposition Left party, in exchange for higher pensions for many Swedes. She also secured the support of coalition partner the Greens.
Of the 349 members of the Riksdag, 174 voted against her. But on top of the 117 MPs who backed Ms Andersson, a further 57 abstained, giving her victory by a single vote.
A former junior swimming champion from the university city of Uppsala, she began her political career in 1996 as political adviser to then-Prime Minister Goran Persson. She has spent the past seven years as finance minister.
Before MPs backed Magdalena Andersson, Sweden was the only Nordic state never to have a woman as PM.