A losing bonus point meant that the Bulls overtook the Stormers as South African Conference leaders, but they would have expected far more after arriving at Eden Park as firm favourites.
The Blues have struggled to find consistency for much of their Super Rugby campaign, results which piled pressure on coach John Kirwan, but a try in each half underpinned an outstanding result.
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.