The win assured the Blue Bulls of a place in the quarter-finals, but they will have to wait to find out their final placing in the group as they lead the third-placed Pumas by three points with the Pumas having one match still to play.
The visiting Bulls ran in eight tries to four against the outgunned Griffons, who managed just one win in their six matches this season.
The Blue Bulls opened the scoring as early as the second minute when flank Jean Cook barrelled his way over for a converted try.
The visitors roared into a 21-0 lead after just 27 minutes, courtesy of tries from left wing Courtnall Skosan and inside centre Jan Serfontein.
The Griffons finally opened their account in the 29th minute when eighthman Nicky Steyn went over for a try that was converted by replacement flyhalf Hansie Graaf.
Graaf had a solid outing and ended with 13 points courtesy of a try and four conversions.
However, the Blue Bulls reasserted their authority thanks to a try from outside centre Ulrich Beyers eight minutes before half-time as the visitors went to the break holding a formidable 28-7 lead.
The visitors continued to enjoy success in the second half and tries from replacement flyhalf Willie du Plessis, right wing Sampie Mastriet, replacement fullback Jerome Pretorius and replacement lock Franco Mostert sealed the victory.
The Griffons managed the consolation of a bonus point for scoring four tries thanks to three second-half tries from Graaf, replacement centre Japie Nel and loosehead prop Danie van der Merwe.
Scorers:
Griffons:
Tries: Nicky Steyn, Hansie Graaf, Japie Nel, Danie van der Merwe; Conversions: Graaf (4).
Blue Bulls:
Tries: Jean Cook, Courtnall Skosan, Jan Serfontein, Ulrich Beyers, Willie du Plessis, Sampie Mastriet, Jerome Pretorius, Franco Mostert; Conversions: Wesley Dunlop (4), Du Plessis.