A repeat victory in front of a sell-out crowd at Vicarage Road would secure the Ospreys a home semi-final at the Millennium Stadium against either Gloucester or Munster, neither of whom are currently setting the rugby world alight.
This year's Heineken Cup final is also being played in Cardiff and Greenwood said: "As soon as the draw came out I was like 'where is my local bookie?'
"After the drubbing the Ospreys gave Saracens two weeks ago they are the overwhelming favourites to win this game.
"The momentum, the confidence and the belief is all with one team."
Welsh rugby is in buoyant mood - Cardiff Blues are also in quarter-final action this weekend - and the turnaround could be traced back to the Ospreys' Heineken Cup pool victory over Gloucester.
The Wales squad, packed with men from the Liberty Stadium, used that 32-15 triumph as inspiration ahead of their Six Nations victory over England at Twickenham.
Wales went on to win the Grand Slam and now the Ospreys are reaping the benefits of that national success, with the likes of Shane Williams and Gavin Henson playing the best rugby of their careers.
Williams was voted the official player of the Six Nations but if Wales had an unsung hero it was Henson, whose rugby has been revitalised since Warren Gatland and Shaun Edwards took charge of the national team.
Henson was defence captain in a side which conceded just two tries all tournament, he tackled like a trojan and revelled in his role as facilitator rather than finisher.
It is not often headline-grabbing stuff, but then Henson has had enough of those.
"I don't particularly enjoy scoring tries as much as the other boys do," Henson said during the tournament. "I would rather put somebody in for a try, I get more satisfaction from that."
Henson's career stalled for nearly three years due to a combination of injury, lost form and a bemusing lack of faith in his ability from the previous coaching regime.
"I don't think I was liked," he reflected.
But Greenwood played with Henson on the 2005 Lions tour and always knew there was a world class player desperate to be let loose. Gatland, Edwards and Ospreys coach Lyn Jones have found the keys.
"I have always, always said he is a top class player. Some people you can just tell have an innate ability to turn a game, go up a level and be world class," said Greenwood.
"Whatever issues there were off the field, he no longer has. He has answered all the questions thrown at him this year.
"He has turned his defence into a huge strength, he has simplified his game. He runs hard, he runs straight and when the opportunity comes to put the ball through hands he does.
"He has increased his work rate, he gets himself muddy and gets in amongst it. He is enjoying the responsibility of being defensive leader.
"Gavin is a quiet guy. He is not someone who wants the microphone, to shout and scream and dance. He just gets on with things on the rugby field.
"You have to allow him to do it his way and that is what the current Wales management team have done. And that is what they allow him to do at the Ospreys.
"It is a problem for Saracens. You look through that Ospreys team and there are so many individuals they have to stop."