John McEnroe's team now require one win from their four matches on Sunday to reach the 13-point mark and claim their second crown, but with each victory worth three points on the final day Team Europe still have some hope.
Team World arrived at Vancouver's Rogers Arena with a 4-0 lead after four wins on the first day of the competition.
With each victory worth two points on the second day, world number eight Fritz put Team World in a commanding position after a thrilling 6-2 7-6(3) win over Russian Andrey Rublev.
"Any type of team environment, I feel it always elevates my game," said Fritz, who avenged his Monte Carlo semi-final defeat by Rublev earlier this year.
"I feel like my record in team events is strong because I have a team cheering for me. I get pumped up. I'm excited to play for them. It just adds more pressure and fire to it."
Casper Ruud pulled a win back for Bjorn Borg's Team Europe with a 7-6(6) 6-2 victory over American Tommy Paul, but Tiafoe defeated Poland's Hubert Hurkacz 7-5 6-3 to give his team an 8-2 lead.
Tiafoe pulled off one of the shots of the tournament with a volley behind his back and followed it up with a winner at the net in the second set to delight the crowd.
"I came out here wanting to have fun," Tiafoe said. "Being in the team environment is so foreign to us. Tennis is such an individual sport.
"I wanted to play well for the guys, for the fans."
Local hero Felix Auger-Aliassime teamed up with American Ben Shelton for a 7-5 6-4 victory over Hurkacz and Gael Monfils to cap off another good day for the defending champions.
