Top seeds Quins are among three English clubs through to the tournament's last-eight, being joined by Saracens and Leicester.
While Quins will host Munster in early April, Saracens entertain Ulster and Leicester go to Toulon, with the remaining quarter-final an all-French affair between Clermont Auvergne and Montpellier.
It is the first time since 2009 that three English clubs have made the quarter-finals.
Leicester progressed as sixth seeds after knocking out four-time European champions Toulouse, which means a trip to Jonny Wilkinson's Toulon.
"You would prefer to be at home, but we have got to get past Toulon. They are an immense side, a fantastic team," Leicester rugby director Richard Cockerill said.
"It is going to be a huge event for us and another huge day in Tigers' history."
While Leicester toppled Toulouse 9-5 in the Welford Road snow, Saracens skated across a wintry surface on their final appearance at Vicarage Road to post a 40-7 bonus point victory over Edinburgh.
And Munster, Heineken Cup winners in 2006 and 2008, knew they had to record a bonus point success against Racing to progress as eighth seeds at holders Leinster's expense, and they duly delivered with a 29-6 win.
"If we hadn't (reached the quarter-finals) the people would be in the long grass, there would have been a lot of bullets being fired," Munster head coach Rob Penney said.
"If we hadn't got the outcome, I would have still been proud of the progress we are making and the areas of growth we are getting."
Clermont, unbeaten at home for more than 50 games, will be firm favourites to eliminate quarter-final debutants Montpellier, while three English clubs secured Amlin Challenge Cup quarter-final home ties.
Top seeds Bath entertain Stade Francais, with Gloucester hosting Biarritz and Wasps meeting Leinster. The fourth tie sees French heavyweights Perpignan and Toulouse going head to head.
Assessing the prospect of Leinster's visit, Wasps rugby director David Young said: "You could look at it and say it's not much reward for topping our pool to then be given a draw against the Heineken Cup champions.
"But but on the flip side, what a hugely exciting prospect it will be playing them at home. It is a game which, if we can win it, would really put us on the map."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.