Zlatan Ibrahimovic scored his first goal for Sweden since 2005 with a stunning 25-yard strike after 67 minutes.
Petter Hansson scrambled home Sweden's second six minutes later to end Greece's dogged defensive resistance.
Hansson had almost given Greece the lead earlier, heading just wide of his own goal, but Sweden deserved the win.
Sweden manager Lars Lagerback lavished praise on Ibrahimovic: "It was very good play before and the conclusion was world class. It was very beautiful.
"The players did an excellent job - it's obvious that Greece are a very difficult team to play against."
Prior to the game Lagerback had warned his side that Greece would prove a tough team to overcome.
"Physically they work very hard, they move their line forward and we worked hard on that," said Lagerback.
"It's fantastic to win, particularly with the way we played, it was a 100% deserved victory.
"We knew they might play up to five defenders and they would have man-markers.
"It's tough to face them because they are physically strong. I'm not surprised they had one marker on Zlatan."