The Quins now have to overcome the Sharks to avoid the drop after being well beaten by the Tykes, who shrugged off the loss of injured Lion Iain Balshaw to take a huge step towards preserving their own Premiership status.
England centre Greenwood showed his class early on with one thundering tackle on winger Andre Snyman and a great break which led to a penalty for his side.
But the Tykes went ahead when Snyman went over in the corner for the game's first try to give Leeds a 7-3 half-time lead.
The Quins hit back after the break when Jeremy Staunton intercepted Gordon Ross' pass and sprinted 50 metres to the line.
However, two late tries from Phil Christophers and Snyman secured the points for the Tykes.