Romania are 24 places below their opponents in the world rankings and had warmed up for the Euros with disappointing goalless draws against Bulgaria and Liechtenstein.
But cheered on by a wall of yellow in Munich, they were gifted the opener when Lunin’s poor clearance was intercepted by Man, who teed up Stanciu for a wonderful whipped first-time shot into the net.
Buoyed by their goal, Romania dominated and Stanciu almost scored direct from a corner minutes later with another curling effort beating Lunin and cannoning off the crossbar.
Real Madrid goalkeeper Lunin had a poor game and was also at fault for Romania’s second when he misjudged the flight of Marin’s strike and the ball went under his attempted save.
Edward Iordanescu’s side added a third to confirm their biggest win in the tournament as Dragus finished following a mazy run from lively Parma forward Man.
Romania – who were unbeaten in qualifying – are now in a strong spot to progress from Group E and face Belgium in their next match on Saturday.
Ukraine, quarter-finalists at the Euros three years ago, disappointed, taking 77 minutes to record a shot on target.
Substitute Roman Yaremchuk struck the top of the crossbar for Serhiy Rebrov’s team deep into second-half stoppage time.
Ukraine face Slovakia in their next fixture on Friday.