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That at least keeps the club's European dreams alive and Edwards and co will now be trying to replicate their success in lifting the title back in 2003.
"There is definitely a lot more kudos in the Amlin Challenge Cup these days with the Heineken Cup teams coming into the quarter-finals. I hear Cardiff is a pretty good party town, so it would be nice to get there for the final in May," said Edwards.
If Wasps are going to reach the Welsh capital they will have to do it the hard way. Not only do they have to travel to face Quins in the last eight, but they also have to go away in the semi-finals if they win.
Awaiting the victors of the all-England quarter-final will be one of two former Heineken Cup winners, Brive and Munster.
"We are really getting to know Harlequins well this season and we are playing them next weekend," said Wasps director of rugby Tony Hanks.
"European rugby is really cut throat and looking at the teams in the Amlin quarters it could easily be the Heineken Cup. If we manage to get past Harlequins then we would go on to face either Brive or Munster.
"We love European rugby at Wasps and we have players here who have won both the Heineken Cup and the Amlin Challenge Cup. Big days and games like this one against Toulouse are why you play rugby.
"I'm pleased we responded so well after the defeat in Glasgow and it showed we are a proud team. But there were mixed feelings at the end of the game.
"We were delighted to have beaten a very good Toulouse side, but still upset when thinking about what might have been had we won in Scotland."
