Bryant made the play in the bottom of the seventh inning of the 12-7 setback Wednesday at Great American Ball Park in Cincinnati.
The Reds led 12-5 when outfielder Shogo Akiyama swung and hit an 0-2 fastball off Cubs pitcher Duane Underwood with the bases loaded.
Akiyama's shot scorched down the third base line, and Bryant dove for the ball and appeared to make an acrobatic catch. He then jumped to his feet and tagged third base for the second out before he rifled a toss over the pitcher's mound to get Akiyama out at first base.
A replay later showed that the ball hit the ground before it found Bryant's glove, but the third-base umpire had ruled that Bryant made the catch.
Reds manager David Bell argued the call, but line-drive infield outs are not reviewable and cannot be challenged by managers.
The Cubs joked about the play after the game on social media. The team's official Twitter account posted an edited version of the Bryant play, which skipped the split-second footage of when the ball hit the ground and bounced into Bryant's glove.
The Reds' Twitter account responded with a photo of an asterisk.
"I was convinced [I caught it]," Bryant told reporters. "I just watched the video now. I don't know. It's kind of hard to tell. I guess I can put that on my list of things that I've accomplished.
"I don't think I've ever turned a triple play in my life. So that's pretty cool."
The Cubs hadn't made a triple play since May 10, 1997.
The 2019 season featured three triple plays. MLB players have completed 41 triple plays since the start of the 2010 season. The 2016 and 2017 seasons each had seven triple plays, the most since 11 were completed in 1979.
Bryant also went 1 for 5 with two RBIs in Wednesday's loss. He is hitting .091 this season. Mike Moustakas, Nick Castellanos and Nick Senzel each homered in the Reds' win.
The Cubs (4-2) will face the Reds (2-4) at 6:10 p.m. EDT Thursday in Cincinnati.