Tiafoe had been 0-11 against opponents ranked in the Top 5.
"That guy is special. He's going to do a lot of great things. Win a ton of Grand Slams," Tiafoe said about Tsitsipas, then broke into a wide smile and added: "But not today."
In the first women's match completed, Aryna Sabalenka — who is seeded No. 2 after defending champion Simona Halep and four-time major champ Naomi Osaka pulled out — advanced with a 6-1 6-4 victory over qualifier Monica Niculescu. Other winners included 2017 champion Garbiñe Muguruza, No. 23 Madison Keys and No. 32 Ekaterina Alexandrova.
The tournament referee began announcing postponements at about 3pm, including matches involving five-time Wimbledon champion Venus Williams, French Open semifinalist Maria Sakkari and seeded men Denis Shapovalov and Alex de Minaur (who is supposed to face up-and-coming American Sebastian Korda).
Still, at least there was tennis being played at the venue that hosts the oldest Grand Slam tournament.
"It feels great seeing everyone and being back at probably the most special, most sacred, tennis court in the world," the top-seeded Djokovic said.
"Alongside many other players, I was very sad last year that Wimbledon was canceled," he said, referring to the first time it wasn't contested since World War II.
"They were very difficult times for everyone, but I'm really glad that the sport is back. Hopefully you guys have enjoyed it and you will enjoy it in the next couple weeks."