Keane, who has previously scored at this level just once before (against San Marino), notched the game's opening goal on 21 minutes when he headed home James Ward-Prowse's free-kick from close range.
The Young Lions then added another goal before the break through Saido Berahino, and the West Bromwich Albion forward sealed the win in the final minute when he finished a move started by young United midfielder Jesse Lingard.
The victory, against a Finland side who finished the game with 10 men following Patrick O'Shaugnessy's dismissal for a second yellow card on 70 minutes, keeps England U21s in pole position to qualify for the 2015 European Championship. Gareth Southgate's men now have 13 points from five games and host San Marino next Tuesday 19 November.
Elsewhere on Thursday night, Marouane Fellaini played 90 minutes and hit the post for Belgium in a 2-0 home defeat against Colombia. The match had been billed as one of the most exciting international fixtures this week given both teams' presence in the top five of FIFA's world rankings and it was the South American side that triumphed thanks to second-half goals from Radamel Falcao and Victor Ibarbo.
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