The Olympiakos Piraeus striker volleyed Greece in front after 14 minutes but the visitors hit back four minutes later via Bogdan Stancu's header from a clever Gabriel Torje free-kick.
The Greeks retook the lead immediately with a Dimitris Salpingidis close-range finish, while Mitroglou added a third in the 66th to increase the advantage. Romania's Costin Lazar was then sent off in stoppage time.
Fernando Santos' team were well worth the victory that gave them every chance of securing a place in Brazil next year after the return leg against Victor Piturca's Romania in Bucharest next Tuesday.
After a frantic opening spell full of errors and niggly fouls, it was Greece who settled first.
Mitroglou, who has scored 19 goals for club and country this season, beat the offside trap to latch on to a superb lofted through ball from Salpingidis and volley home past Romania's veteran keeper Bogdan Lobont.
It was the start of a frantic six minutes, with Torje's whipped free-kick nodded in by Stancu at the back post and Greece then going ahead again with the Romanian celebrations barely over.
Some slack marking from the visitors allowed Celtic's Giorgos Samaras to pick out right back Vassilis Torosidis and his first-time cross was slid home from close range by Salpingidis.
Samaras had the chance to put Greece 3-1 up when he burst into the box in a counter-attacked started by defender Sokratis Papastathopoulos, but the tall striker shot straight at Lobont.
Torje presented Romania's main attacking threat with penetrating runs down the right channel but it was Razvan Cocis who missed a good chance when he blazed just over the bar after finding space inside the area.
Neither side could maintain the tempo of the first half after the break but it was the irrepressible Mitroglou who provided the cutting edge, rifling home Kostas Katsouranis' flick-on from a Samaras free-kick after 66 minutes.
Captain Katsouranis, who was booked and will now miss the second leg, and Mitroglou both missed further chances with the latter having a goal-bound effort blocked by Lobont to deny him what would have been a fifth hat-trick of the season.
(Reporting by Graham Wood, editing by Alan Baldwin)
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