Folau played a major role with his ability to win the ball in the air and to draw defenders before passing.
But the Crusaders stayed calm and chipped away at the Waratahs' lead, making use of their almost test-strength forward pack.
The Waratahs paid heavily for losing two players to yellow cards as the Crusaders scored 31 successive points.
The Crusaders scored tries through hooker Codie Taylor and winger Seta Tamanivalu to cut the lead to 29-19 at halftime, then narrowed the margin to five points with a try to winger Braydon Ennor.
They finally took the lead with a penalty try from a series of scrum infringements, edging ahead for the first time in the match with Richie Mo'unga's conversion.
Waratahs flyhalf Bernard Foley missed two crucial penalties in the second half and a late attacking move ended with a knock-on.
"I don't know where we were in that first 20 or 25 minutes," Crusaders captain Sam Whitelock said. "But full credit to the boys, we were nice and calm out there the whole time even though we were down 29 points."
An Australian team did have a win, but only against another Down Under side. The Melbourne Rebels beat the ACT Brumbies 27-24.
The Johannesburg-based Lions, beaten finalists the last two seasons, lost a third straight game on tour in Australia and New Zealand by losing to the Highlanders 39-27 in a thrilling match at Dunedin.
The Highlanders took led 14-0 early with tries to Lima Sopoaga and Liam Coltman but the Lions rallied and had cut the lead to 19-15 at halftime.
The Lions kept up steady pressure throughout the second half and tries from lineout drives allowed them to close to 33-27 after 72 minutes. The Highlanders pulled away with two penalties in the last five minutes.
That left the second-place Lions 11 points off the pace set by overall leader the Crusaders. The Waratahs are third overall as the Australian conference leaders but lag 16 points behind the Crusaders.
Later in South Africa, a try in the first half by Anton Lienert-Brown and a penalty try from a scrum in the second was enough for the Chiefs to win 15-9 and end the Stormers' unbeaten run this season at home in Cape Town.The Chiefs prevailed in a game dominated by errors and ill-discipline from both teams.
The Stormers managed just three penalty kicks.In contrast to that dour affair in Cape Town, the Bulls scored five of the eight tries to beat the Sharks 39-33 in a high-tempo match that swung back and forth.
The Bulls wrestled control for good with lock Lood de Jager's try with 15 minutes to go.Down to 14 men because of center Lukhanyo Am's yellow card, the Sharks scored their third try through flanker Jean-Luc du Preez straight after Am's exit.
His brother Robert kicked a penalty after the final hooter to take a losing bonus point and leave three teams — the Bulls, Sharks and Jaguares — tied for second on points in the South African conference.
