Played in front of a bumper crowd at a sodden FMG Stadium Waikato, the Chiefs stayed in contact through the boot of Damian McKenzie before Mark Telea extended the Blues to an unassailable 24-12 lead in the 72nd minute.
Rising youngster Hoskins Sotutu played a leading hand in the victory. The powerful 21-year-old scored the Blues first try and then laid on a superb pass to put Telea over late in the game to showcase his wide array of skills.
The win also marked a club record of five wins away from home for the Blues.
Next week the Blues face the Highlanders at Eden Park on Saturday night while the Chiefs travel to Christchurch for a showdown with the Crusaders on Sunday afternoon.
Chiefs: 12 (Damian McKenzie 4 pen) Blues 24 (Hoskins Sotutu, Mark Telea tries, Otere Black con, 2 pen, Beauden Barrett dg, pen)