Fergus Burke, Will Jordan, Brodie McAlister and Tom Christie scored the tries for the Crusaders on Friday, who will host the seventh-placed Reds for a second successive week in the quarterfinals.
The Crusaders were some way off their machine-like best but shot out to an early 14-0 lead with a two-try burst in four minutes.
Flyhalf Burke was first to cross after the Reds botched a scrum deep in defence, with returned fullback Will Jordan touching down in the 14th minute after a turnover in midfield.
The Reds stemmed the bleeding for about 20 minutes and clawed back three points, with Lawson Creighton knocking over a penalty.
But the Crusaders went to halftime with a commanding 21-3 lead after hooker McAlister rumbled over from a rolling maul for their third try three minutes before the break.
With Crusaders loose forward Pablo Matera yellow-carded after the break for a lifting tackle, the Reds rallied through a try from Richie Asiata in the 57th minute.
But Christie sealed the win with the Crusaders' fourth try 15 minutes later before the Reds' Harry Wilson grabbed a consolation try a few minutes from the siren.