Tosun is top scorer of the Turkish league but the goals were the first he has registered for the national team for exactly one year.
On nine minutes, Ismail Koybasi whipped in a cross from the left, which Barcelona midfielder Arda Turan flicked on for Tosun to net.
Four minutes later Tosun produced a looping header from Selcuk Inan's corner to beat goalkeeper Lukas Hradecky at the far post.
The win moves Turkey above Iceland in third place on eight points, behind Ukraine on goal difference having played one game more. Finland remain fifth with one point from five games.
(This version of the story corrects location of match in first paragraph)
(Reporting by Ece Toksabay, editing by Neil Robinson)
