After an uncharacteristically slow start, the Crusaders hit their straps in the second half after leading 11-10 at the break.
Last year's champions scored four second-half tries, dominating possession and restricting the Chiefs to a consolation 72nd minute try to winger Etene Nanai-Seturo.
The Chiefs had started well, Damian McKenzie's try coming after just three minutes on the back of a strong run after the Chiefs moved the ball wide with purpose and precision.
McKenzie's accuracy with the boot added the conversion, and a penalty six minutes later to give the visitors a 10-0 lead inside 10 minutes.
It took the Crusaders a while to find their rhythm, but winger Leicester Fainga'anuku got them on the board with a spectacular diving finish after a David Havili break.
Two Richie Mo'unga penalties gave the Crusaders the narrowest of halftime leads, which was extended four minutes after the restart by a penalty try.
With halfback Brad Weber yellow-carded for an offside infringement, the Chiefs were under the pump but a strong defensive effort held the Crusaders at bay until just before his return.
Will Jordan scored between the posts after a strong Fainga'anuku carry and quick ball movement created gaps,
Tries to Whetukamokamo Douglas and Mitchell Dunshea late in the half wrapped up the game to give the Crusaders their third win from as many games this season.