Felix Auger-Aliassime and Ben Shelton added a doubles win for Team World, who take a 10-2 lead into the final day.
With each match win worth three points on Sunday, Team World will need just one victory to lift the trophy for a second time - after Europe claimed the first four editions of the event that began in 2017.
"We're just going to keep doing what we're doing," Team World captain John McEnroe said. "Felix said the job's not done - but we're pretty close."
Tiafoe had lost his last three matches against Poland's Hurkacz, but the 11th-ranked American turned the tables with a 7-5, 6-3 victory that thwarted Team Europe's hopes of building momentum in the wake of Casper Ruud's 7-6(6), 6-2 victory over Tommy Paul.
Ruud's win gave Europe their first points of the event, after Team World swept Friday's four matches and Fritz opened day two with a 6-2, 7-6(3) victory over Europe's highest-ranked player, world number six Andrey Rublev.
Norway's Ruud, ranked ninth in the world, admitted he felt some pressure taking the court with his team trailing 6-0 - and he acknowledged that he got away with "a bit of a steal" in the first-set tiebreaker, saving a set point at 6-5 before pocketing the set.
He was dominant in the second set, however, breaking Paul to open the set and breaking the American again with a blistering service return for a 5-2 lead.
Ruud served it out with a love game, but his hopes that it would be "a momentum shift" were dashed when Tiafoe opened the night session at Rogers Arena with a crowd-pleasing win over Hurkacz.
World number 16 Hurkacz got off to a strong start and had a set point against Tiafoe's serve at 5-4 in the first.
Tiafoe worked out of the jam, then grabbed the first break of the match in the next game before sealing the set with a hold at love.
Having broken Hurkacz to open the second set, Tiafoe was freewheeling, delighting fans in the penultimate game with a behind-the-back shot to keep alive a rally that he won with a backhand volley.
"We all get lucky sometimes," said Tiafoe, who broke at love to seal the win, Hurkacz producing his only double-fault of the contest on match point.
Staying with it
Hurkacz was immediately back in action, teaming with France's Gael Monfils in the closing doubles won by Auger-Aliassime and Shelton 7-5, 6-4.
Eighth-ranked American Fritz had opened the day's action, rallying from a break down in the second set to beat Rublev.
Fritz's aggressive ground strokes were much in evidence, but a pair of deft half-volleys - one to give himself a game-point for 6-6 and another in the tiebreaker - also stood out.
Fritz had opened the match with a service break and cruised through the first set. He said Rublev's early break on the way to a 3-0 lead in the second did nothing to shake his confidence.
"He just played a really good game," Fritz said. "I just told myself if I stayed with it then I could find a chance to get the break back."
After one point was awarded for each win on day one, each win on Saturday was worth two and it will be three on Sunday, when the competition opens with a doubles clash followed by three singles matches.
