© Getty Images
The home side had the first opening of the game when Shaun Williams was afforded space and time just outside the area, but he failed to hit the target as he blazed his strike over the bar.
Middlesbrough thought they had taken the lead as Vossen powered a header at goal which looked to have crossed the line before David Forde clawed it away, but the officials ruled that it was not a goal for the forward.
However, Vossen did fire his side ahead in the 21st minute when he latched on to Daniel Ayala's ball over the top of the defence and the striker maintained his composure as he slotted the ball past Forde.
Middlesbrough doubled their lead minutes later when Vossen's cross was deflected into the path of Patrick Bamford at the back post and he slotted the ball into the bottom corner.
Vossen continued his excellent first half when he notched his second goal of the game as he completed a fine attacking move from the visitors by poking the ball past Forde from close range.
© Getty Images
In the second half, Bamford nearly added a fifth goal for Aitor Karanka's men as he pounced onto a loose ball in the penalty area, but he somehow missed the target from close range.
Bamford went close again as Albert Adomah did well down the right to carve out the opportunity for the striker, but he lost his composure at the vital moment as he fired over the bar.
Millwall scored a consolation goal late in the game as Scott McDonald netted against his former club as he turned the ball into the net from close range.
However, Middlesbrough restored their four-goal advantage a minute later as Kike collected the ball and smashed an effort off the inside of the post and into the net as the visitors coasted to victory.