The seven-time defending champions fought back from 10 points down in Christchurch to hand the Blues their second loss of the season.
It was only the third win for the Crusaders, whose playoff hopes would have ended with defeat.
Instead, they improve to ninth and can still sneak inside the top eight if they beat Moana Pasifika in the final round next week and other games fall in their favour.
All hope seemed lost when the Blues surged ahead 22-12 soon after half-time through a converted try to winger AJ Lam.
The home side regrouped and crossed through Christian Lio-Willie and Noah Hotham to reclaim the lead and hold it to the end, even after conceding a second try to Lam.
Scrum-half Hotham, a standout performer, said the Crusaders lifted their standards
"This means so much. It was a big week for us, we haven't put in the performances we wanted to over the last couple of weeks," Hotham said.
"In our last games, we've dropped off on those last 20 minutes so it was just trying to learn lessons and play the full 80."
A bonus point leaves the Blues level on top of the standings with the Wellington Hurricanes but still in first place because of a superior points differential.
Blues captain Patrick Tuipulotu said they need to improve their discipline after a yellow card to fullback Stephen Perofeta swung the contest in favour of the Crusaders.
"We were on the back foot and they had a lot of the momentum in that second half," Tuipulotu said.
"They made us pay, playing with one man down.
"Throughout the game, there was a lot of ill-discipline, so that's very disappointing for us."
Prop Ofa Tu'ungafasi scored two first-half tries to help put the Blues 15-12 ahead at the break, with Ethan Blackadder and Chay Fihaki having crossed for the Crusaders.
Earlier, Moana Pasifika beat New South Wales Waratahs 27-12 in Auckland to knock the last-placed Australian side out of playoff contention.
Winger Fine Inisi scored a double as the Pacific Island side ran in the first four tries inside 60 minutes to lead 27-0.
Veteran former Wallabies prop Sekope Kepu was among the try-scorers, one day after announcing he would retire at the end of the Super Rugby season.
