A hat-trick of Rob Miller penalties saw Sale over the line at the Parc des Sports Aguiléra, with the full back's final strike just two minutes from time handing the visitors a 9-7 victory and a home tie in the quarter-finals.
And to make a good week even better, Sale have announced that Scotland second row Hines will be moving to Manchester at the end of the season.
"Nathan is a world class player and will fit in well with us at Sale Sharks," said Sale director of rugby Steve Diamond, who highlighted the 37-year-old's form in the Heineken Cup ahead of his summer switch to the AJ Bell Stadium.
"He has vast experience playing at the highest level and is a regular starter at Clermont. Last weekend he was magnificent in helping his team beat Racing Metro.
"The forwards at Sale Sharks have played well this season and adding Nathan to the group will be a big boost to us."
As well as winning 77 caps for Scotland and representing the British & Irish Lions on their 2009 tour of South Africa, Hines has played 83 times in the Heineken Cup, with all but five of those appearances coming in the starting XV.
The Australian native claimed the continent's greatest prize with Leinster in 2011, starting all nine games as the Irish outfit won a second Heineken Cup crown. He has also played in the tournament for Edinburgh and Perpignan and picked up a runners' up medal with current club Clermont last May when they lost out to Toulon in the Dublin showpiece.
Hines has missed just one Heineken Cup in match the past five seasons and has been involved in all six ties this term as Clermont eased to top spot in Pool 4 and booked a home quarter-final with Leicester Tigers at the Stade Marcel Michelin.