Hallur Hansson put the Faroes ahead after 32 minutes and Brandur Olsen added a second goal in the 70th before Sokratis Papastathopoulos reduced the deficit six minutes from time.
Greece nearly scored in the first half as Nikos Karelis's shot was blocked on the line and Kostas Mitroglou hit the woodwork just before the Faroes scored on the break when Hansson turned in Solvi Vatnhamar's cross.
In the second half, Faroes keeper Gunnar Nielsen turned a Karelis drive on to the post and Vasilis Torosidis thought he had scored but his effort was disallowed for a foul.
Greece pushed forward for an equaliser but conceded again when Solvi Vatnhamar set up Olsen to shoot past Orestis Karnezis. Papastathopoulos headed in a corner late on to reduce the arrears but the hosts held on.
The Faroes are now fourth with six points, five points behind third-placed Hungary who earlier won 1-0 in Finland. Romania lead the group with 14 points ahead of Northern Ireland on 13 after the top two sides drew 0-0 in Belfast.
(Writing by Ken Ferris; editing by Ed Osmond)
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