"We say at the start of every season that the measure of success is whether or not you are in Europe," said head coach Edwards.
"We have overachieved in recent seasons and we need to refocus now on climbing up the table and guaranteeing ourselves European rugby next season.
"It's not going to be easy, we are going to lose some players on international duty, but we have prided ourselves on our mental strength in recent years. We need some of that strength of character now.
"The reason we won the Heineken Cup in 2004 and 2007 was because of our clinical finishing.
"That just wasn't the case against Castres and it cost us dear. We are very devastated."
Webber's 52nd-minute mistake will live long in his memory as he somehow failed to gather the ball off the floor with the line at his mercy.
Although Waldouck followed up Danny Cipriani's superb first-half try with another solo score to put Wasps ahead on 68 minutes, there was always a sense the French would hit back.
Van Gisbergen had a chance to kill the game off but couldn't collect Dave Walder's pass following a sharp break by the replacement fly half, and moments later Castres full-back Thomas Bouquie scored his second try.
Taking advantage of Paul Sackey lying injured on the floor and Josh Lewsey's decision to fly out of the defensive line, Bouquie galloped past Van Gisbergen and Flutey for the decisive score.
Harlequins, meanwhile, guaranteed themselves a home quarter-final against Leinster with a 29-24 win over Scarlets.
