Goffin entered the draw as a lucky loser after squandering two match points to lose his final qualification match against Luca Nardi on Sunday, but he produced a fine display to see off the world number one in one hour and 46 minutes.
The Belgian had to rediscover his composure to take the opener after throwing away a 5-2 lead, but ultimately deserved his straight-sets win over the US Open champion.
Speaking on court after his victory, Goffin said: "I always believe that I have the level to cause some trouble against those guys.
"When you play against the world number one on a big stage, big crowd, the fire inside gives you so much power to play your best tennis because you don't have any choice."
Stefanos Tsitsipas was the only other seed in action in Kazakhstan on Tuesday, and he advanced to the round of 16 by beating home favourite Mikhail Kukushkin 6-3 6-4.
The Tokyo Open also lost its top seed, as Casper Ruud fell to a 6-3 6-3 loss to Jaume Munar, who claimed his first top-10 win since beating Alexander Zverev in 2019.
But the likes of Borna Coric and Nick Kyrgios fared better, claiming straight-sets wins over Thanasi Kokkinakis and Tseng Chun-hsin, respectively.
Wimbledon runner-up Kyrgios only dropped four further points on his serve after being forced to save a break point in his opening service game, racing to a dominant 6-3 6-1 win.
Elsewhere, eighth seed Dan Evans fought back to beat Radu Albot 6-7 (3-7) 6-1 6-4, and Alex de Minaur lost 6-3 6-2 to Kwon Soon-woo.
American Kovacevic was only playing in the first round as a lucky loser in Seoul after he was beaten in the final round of qualifying, but he grasped his chance by winning 6-4 6-4.
The 24-year-old served 14 aces and broke the world number 32 three times to set up a second-round meeting with Christopher O'Connell or Tseng Chun-hsin.
Radu Albot got the better of Hiroki Moriya in his first appearance in Seoul, while Jaume Munar progressed when Yosuke Watanuki retired early in their match due to an ankle injury.
Kaichi Uchida came out on top in the final match of the day, consigning Hong Seong-chan to a 6-4 6-2 defeat.
Murray had been due to play Pablo Carreno Busta in the second round in California, only for the 15th seed to pull out with injury.
The three-time major champion instead defeated Albot in 6-4 6-3 for his first straight-sets victory of the season.
That sets Murray up to play fellow Briton Jack Draper in round three ahead of a potential round-four meeting with top seed Carlos Alcaraz, the best player in the draw after Novak Djokovic was unable to enter.
Murray was in complete control against Albot but played with nerves, explaining: "I didn't know if it was because of [the change of opponent].
"I was trying to give reasons for why I may have felt that way. I felt nervous on the court.
"A lot of the way that Radu plays and Carreno Busta plays is not too dissimilar, in terms of the way they like to play points and maybe where and how they like to serve.
"Obviously there are differences in their games, but it wasn't a huge change in terms of my game plan or what I was expecting going into the match.
"But maybe I felt like it was a good opportunity for me and that the draw had opened up slightly. I obviously wanted to try to finish the match quickly."
Yet those nerves are not a problem for Murray, who says he would rather play under that pressure.
"I generally much prefer feeling nervous when I play matches and feeling pressure," he said. "It just felt a bit out of sync with the situation. I was in, I guess, pretty decent control of the match.
"I was serving well, I was hitting the ball pretty well, but there was just more tension than probably there should have been out there on the court.
"But usually I prefer to feel nerves, providing they are in check. And maybe they weren't, for whatever reason.
"I would way rather feel more tension than feel flat, which is really never a good sign for me. If I wake up and I don't feel any of the butterflies or anything like that, it's generally a bad sign."
The 2021 Wimbledon semi-finalist needed 109 minutes to claim a 7-5 6-4 win over Munar, assuming control by breaking in the seventh game of the opener before doing likewise early in the second set.
Canadian Shapovalov – who is seeded fourth for the event – had not featured since a dramatic loss to Andrey Rublev at the US Open earlier this month, but produced a consistent display in Seoul, not committing a single double fault during the match.
His reward for the victory is a last-eight tie with Radu Albot, who held his nerve in two tie-breaks to record a 7-6 (7-3) 7-6 (7-3) win over Steve Johnson.
In Wednesday's other match, Japan's Taro Daniel beat Ecuador's Emilio Gomez in straight sets in the duo's round-of-32 tie.
Top seed Casper Ruud begins his campaign against Nicolas Jarry on Thursday.
Djokovic, who is hunting his first grand slam title of the year on the back of claiming Olympic gold in Paris, won 6-2 6-2 6-4 on Sunday.
However, Djokovic was not at his best, making 40 unforced errors and 10 double faults.
"I wanted to kick start the tournament in the right way and I think I did," Djokovic told ESPN.
"Some ups and downs which I think is normal, getting the rust off your shoulders coming off a different surface and the Olympic Games.
"I haven't played on hard courts for six months, so I'm still finding that groove, finding the tempo on the court."
Next up for Djokovic, who would claim his 100th Tour-level title should he triumph in New York, is his compatriot Laslo Djere, who defeated Jan-Lennard Struff.
Data Debrief: Djokovic equals Federer
Only the great Jimmy Connors (98) has won more US Open singles matches than Djokovic, who claimed his 89th victory at Flushing Meadows to match Roger Federer's tally.
This was also Djokovic's 78th win at Arthur Ashe Stadium, a competition record for a male player.
