Tiafoe, seeking to become the first American man to win the trophy since Pete Sampras bagged his seventh title at the major in 2000, recovered after dropping his serve early in the first set to level at 2-2 before rain halted play.
"It was tough but... it's Wimbledon, you've got to expect stuff like that. I'm happy to get the win and hope you guys enjoyed it," said Tiafoe.
The players returned but were back inside two games later and when play resumed again, it was Tiafoe who took charge of the tight opening set in the tiebreak and closed it out with a powerful serve.
"I love playing on grass and I have high hopes here for Wimbledon," added Tiafoe, who warmed up by winning the Stuttgart grass court title.
"I definitely didn't come along for a vacation. I want to do real damage here and hopefully be here two Sundays from now."
Wu, making his Wimbledon debut and his fourth appearance at a Grand Slam, took a medical timeout early in the second set and had his pulse checked but resumed following a lengthy off-court evaluation.
The world number 62 did not seem too badly affected by the unspecified issue but Tiafoe shifted gears to double his lead in the match and after freely trading breaks in the third, closed out the victory with a solid service game.
"Obviously, he's a high quality player. It's tough, he wasn't feeling well. With that and all the rain delays there was a lot of waiting," Tiafoe said.
"I'm a really relaxed guy but I do like finishing a tennis match. I was happy how I handled it."
The 10th seed, who reached the fourth round last year, meets Swiss qualifier Dominic Stricker (20) in the next round.
Last year's quarter-finalist Taylor Fritz (25) also booked a spot in the second round and a clash with Mikael Ymer (24) after a 6-4 2-6 4-6 7-5 6-3 win over Yannick Hanfmann (31).
That match was suspended on Monday due to darkness and could not resume on a rainy Tuesday with Fritz up 3-2 in the decider.
The world number nine, along with Tiafoe, is now looking to become the first American men's Grand Slam champion since Andy Roddick won the US Open in 2003.