Sharks skipper Juan Martin Fernandez Lobbe, 27, scored in the fifth and 23rd minutes before Andy Tuilagi, Eifion Roberts and David Tait added second-half touchdowns to ensure a turbulent week at Edgeley Park ended on a high note.
Saint-Andre announced on Thursday he is to quit the club at the end of the season after five years in charge.
He is poised to return to his native France and has several suitors, most notably Toulon, but his dream to bow out as a European champion with Sale remains alive after this win.
The Sharks head to Montauban for the return fixture next Saturday and Saint-Andre said: "We won, scored five tries and played some good rugby. "We are still in the competition but next week at Montauban will be a different story.
"There would have been no excuses tonight because the news of my announcement did not affect how they played on the pitch."
Sale have already decided that Kingsley Jones will take over as director of rugby in May. Saint-Andre and Jones have worked together for the last four years and guided the club to the Premiership title in May 2006.
Decision
Speaking about his decision to quit the club, Saint-Andre added: "I tried to be very honest with the guys.
"We've had five fabulous years but it's not a time to say thank-you yet because I will be here until the last game of the season."
However, Saint-Andre pointed an accusing finger at Sale's stay-away fans after just 6,962 turned up for the match.
"I was disappointed with the crowd tonight," said the Frenchman. "The guys deserved more than for such a big Heineken Cup game."
Fly-half Charlie Hodgson converted four of his side's five tries and also claimed a penalty in an impressive all-round display.
Montauban's hopes of qualifying are over after they fell to a third straight defeat but coach Laurent Travers said he was content with his side's efforts.
The visitors' only points came from a penalty and a drop-goal by fly-half Cedric Rosalen, but Travers said: "We conceded five tries but I'm proud of the players.
"They were up against a top team from what is a very strong Guinness Premiership.
"Next week we will play a different team and will be going all out to get the win."
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Chabal (07/12/2008 at 18:35)