Having gone into the final 10 minutes with the scores tied at 21-21, the Crusaders twice thought they had scored late tries as Andrew Ellis and Tom Taylor crossed the line, but neither was able to ground the ball to the satisfaction of referee Keith Brown.
But they persevered and Israel Dagg scored the winning try in the 75th minute as the Cheetahs defence was broken down thanks to sustained pressure.
The Crusaders were using Christchurch Stadium as their home for the first time after the loss of AMI Stadium to the 2011 earthquake.
The Cheetahs has been superb in their ability to come back by sheer movement of the ball.
Teenage Cheetahs fly-half Johan Goosen belied his age with a cool, composed performance, not only with his impressive goal-kicking but also in support play and ball distribution.
Hooker Adriaan Strauss was a constant presence in running the ball in close quarter play while flanker Heinrich Brussow battled all night in the loose.
The teams exchanged early penalties as Taylor registered in the sixth and 30th minutes, missing one, while Goosen struck on the quarter-hour mark for the Cheetahs.
It took a piece of dazzling work in the 35th minute to secure the first try for the Crusaders and it came when Ryan Crotty ran back inside on a dummy run which allowed Taylor to get the ball inside to wing Zac Guildford. He lobbed a pass over a defender to centre Robbie Fruean who linked quickly with Dagg to put right wing Sean Maitland down the sideline for the try.
After the break Dan Carter returned to action for the first time since the groin injury which forced him out of the World Cup. His effect was instant and in the 42nd minute he moved the ball out to Fruean who broke free to the line.
But the Cheetahs demonstrated why they have the reputation as the most attacking team in South Africa three minutes later.
Wing Cameron Jacobs robbed lock Luke Romano of the ball, and the ball was moved quickly forward. Willie le Roux was caught by Goosen raced forward in support to cross the line.
While he missed the conversion, he landed a penalty goal four minutes later.
Taylor extended the Crusaders' lead to 10 points with a penalty goal, but the Cheetahs were growing in confidence, and the introduction of loose forward Ashley Johnson gave his side momentum which troubled the home defences.
The Cheetahs' reward came when the Crusaders failed to gather a downfield kick. Full-back Hennie Daniller made the most of the chance to race down the flank, and the ball was moved back across field where replacement three-quarter Rayno Benjamin got through the defence to feed centre Robert Ebersohn to score the try.
When the Crusaders were penalised for offside at a lineout on their 10m line, Goosen stepped up for the 40m attempt which he landed in the 70th minute to level the scores, but there was still time for the Crusaders to secure the winning try at the third attempt.