The visitors went into the match four points off the top of Pool Two, but Cardiff Blues' bonus-point victory over Wasps left them needing a win to stand any chance of progressing.
They got off to a decent enough start when the Falcons conceded a penalty within kicking range after just four minutes, and Joe Ford made no mistake to open the scoring.
Ford kept the scoreboard ticking over for Sale, though, adding two more penalties to edge his side back into the lead midway through the first half.
Clegg could not add the extras on that occasion, but it mattered little as the Falcons went in for a third score shortly afterwards, Uili Kolo'ofai the man to go over this time.
Ford added his fourth kick of the day to make it 17-12, but the hosts ensured that it would be comfortable at the break when Joshua Furno was driven over the line for a bonus-point score right on the stroke of half time.
Things got even tougher for the visitors early in the second half as George Nott was sent to the sin bin, and Clegg took immediate advantage with a penalty to edge his side further ahead.
Another kick followed to open to a 20-point lead, while Smith added his second try of the afternoon from a quick tap penalty to put the result beyond doubt with 15 minutes remaining.
Sale finally got their first try of the match with less than 10 minutes to go as Viliami Fihaki touched down, but it was too little, too late as the Falcons eased to the victory.