Malmsten was born in Stockholm, Sweden. He was the son of Karl Malmsten and Inez Cadier. Malmsten's career took off when he won a competition in 1916 to design furniture for the Stockholm City Hall (stadshus); in 1917, he exhibited alongside well-known Swedish architects such as Gunnar Asplund and Uno Åhrén.

Malmsten founded an eponymous furniture store on the riverfront of Stockholm. The store is currently run by his grandson.

In the 1920s he was invited to furnish a room for the crown prince, Gustaf VI Adolf, and his wife Louise in Ulriksdal Palace.

He was honoured with an inaugural Prince Eugen Medal for design in 1945.