A packed crowd of 32,000 at Karaiskakis Stadium, which included Greek President Karolos Papoulias, saw coach Fernando Santos's squad create a convincing victory which provided a good cushion for the return match in Romania on Tuesday.
Mitroglou scored one goal in each half and PAOK forward Dimitris Salpingidis added another in the first half for the Greeks.
Greece opened the scoring in the 14th minute when 25-year-old Mitroglou slotted the ball past Roma goalkeeper Bogdan Lobont from up close into the right corner of the net.
The play was created by Salpingidis who cleverly lifted the ball past the Romanian defence looking for offside.
The visitors deadlocked the score at 1-1 in the 19th minute when the Greek defence fell asleep and left Stancu all alone at the far right corner and he headed the ball past Granada keeper Orestis Karnezis after a free-kick from the far left by Espanyol midfielder Gabriel Torje.
However a minute later Greece once again took the lead with Celtic forward Giorgos Samaras feeding Roma defender Vasilis Torosidis who sent a perfect cross from the right side to a dashing Salpingidis who hit the net from up close.
The insurance goal for the Greeks came in the 66th minute when Samaras lifted a free-kick from the far left side to PAOK midfielder Kostas Katsouranis who headed the ball to a waiting Mitroglou who tapped the ball into the Romanian net from up close for the 3-1 advantage.
Five minutes later Mitroglou nearly added a hat-trick in the match when he missed a shot from up close as Lobont made a good save.
In the 39th minute Romania had a good chance to level again when a cross from Stancu found Torje in scoring position but his shot went high.
In the first minute of play Mitroglou was brought down in the penalty box by Steaua Bucharest defender Florin Gardos, but Portuguese referee Pedro Proenca saw no reason to blow his whistle.
Three minutes later Getafe forward Ciprian Marica took a shot from up close for Romania only to have it blocked by Karnezis.
In the seventh minute of action a long shot by Torje did not find the mark but had the Greeks concerned about the added pressure on their defence.
Romania midfielder Alexandru Bourceanu of Steaua Bucharest received a yellow card from Proenca in the 86th minute resulting in him missing the return leg with the Greeks on Tuesday.
And Greek captain Katsouranis was also yellow-carded in the 62nd minute and will be missing the match in Romania.