They have only managed two victories on the road this season - against Saracens at Twickenham on the opening day and in the European Cup at Italian minnows Treviso.
But Saturday's efforts - including 17 points from Alex King and a try from Dominic Waldouck - ensured a losing bonus point should be enough.
"It's down to two now isn't it and our away form won't be poor next week," said Wasps director of rugby Ian McGeechan.
British and Irish Lion Dallaglio added: "The prize for whoever goes through next week is huge and we are not a million miles from turning that corner.
"We have more losing bonus points than any other team (apart from Worcester in the Guinness Premiership).
"Beating Perpignan gives us a huge amount of confidence going into next week."
Perpignan captain Perry Freshwater, the England loose-head, expects Wasps to go on and seal a place in the last eight.
"They are one of the best sides in Europe. They are always physically very, very fit and they play a faster game than we are used to and they do it very well," he said.
"They will have gained confidence from today's performance and if they play with the same tempo they will be well placed.
"It is down to one game at the end and going to Castres is never an easy thing to do. But I'd be feeling fairly comfortable if I was in the Wasps team at the moment."
Wasps played down an injury scare surrounding England captain Phil Vickery, who crumpled to the deck after hitting a ruck awkwardly midway through the second half.
Initial fears Vickery had suffered a recurrence of his back injury were allayed when he stood up, albeit groggily, to continue.
Vickery was central to a powerful Wasps scrum and his performance impressed opposite number and England team-mate Freshwater.
"Wasps attacked us in the scrum and it is good to see him playing well. He had a tremendous game in the scrum today," added Freshwater.