Full-back Lolagi Visinia claimed an 18th-minute touchdown, but it was centre George Moala's try midway through the second period that confirmed an outstanding victory.
Fly-half Ihaia West added 13 points via three penalties and two conversions to leave the Blues shell-shocked.
Centre Jan Serfontein touched down twice for the visitors - scoring once in each half - while fly-half Handre Pollard kicked two penalties and a conversion, but the Bulls could have few complaints after being outdone by a team that produced arguably its best performance of the Super Rugby season.
While the Blues will look to build on what was a largely unexpected triumph, the Bulls will have to regroup quickly.
They now face appointments with the Chiefs and Brumbies, so the star-studded South African outfit will have little time to dwell on what was a poor start to their overseas tour.