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.
The Serbian brushed aside Cristian Garin in his opener and did likewise in Thursday's match with Van De Zandschulp, prevailing in just 71 minutes in the Kazakh capital.
Djokovic has now won six matches in a row and is seeking back-to-back titles after triumphing at the Tel Aviv Open last week – his 89th ATP singles crown.
He reeled off five games in a row in the second set to advance 6-3 6-1, with that his 30th tour-level victory of the year.
And as Djokovic nears the end of a rather mixed season, the 21-time grand slam winner is in the mood to add further titles to his collection.
"My season is different from any other so don't look at my season in comparison to others," he said in his post-match interview.
"I haven't played for over three months before Israel so I definitely am physically and mentally motivated to do well."
Awaiting Djokovic in the quarter-final is Karen Khachanov, who eliminated Marin Cilic with a 2-6 6-3 6-3 comeback victory.
World number four Daniil Medvedev also advanced thanks to a 6-3 6-2 win over Emil Ruusuvuori, while Hubert Hurkacz beat Alexander Bublik 6-4 6-4.
At the Japan Open, third seed Taylor Fritz defeated Hiroki Moriya in three sets and will now face fifth seed Nick Kyrgios, who recovered to beat Kamil Majchrzak 3-6 6-2 6-2.
Denis Shapovalov, looking to build on his run to the Seoul Open final, got the better of home hope Rio Noguchi with a 6-3 6-1 win to advance to the quarter-finals.
Having won his 89th tour-level title on Sunday in Tel Aviv without dropping a single set, fourth-seed Djokovic wasted no time in securing a win to set up a second-round clash against Botic van de Zandschulp.
Also in Kazakhstan, third seed Stefanos Tsitsipas made hard work of his second-round tie against Luca Nardi, requiring two tie breaks in a two-hour showdown in a 7-6 (7-2) 7-6 (7-3) victory to secure a spot in the last eight.
Andrey Rublev, Roberto Bautista Agut and Adrian Mannarino also sealed their places in the quarter-finals at the Astana Open on Wednesday, while Marin Cilic and Emil Ruusuvuori won their first-round matches.
Meanwhile, Taylor Fritz defeated James Duckworth 6-2 6-7 6-1 in the Japan Open, where there was also a victory for Denis Shapovalov against Steve Johnson.
In the doubles, top seeds Nick Kyrgios and Thanasi Kokkinakis overcame home duo Yoshihito Nishioka and Kaichi Uchida, with second seeds Matthew Ebden and Max Purcell also progressing past the round of 16 stage.
The two played out a thrilling first two sets, with Medvedev taking the lead before Djokovic fought back to level the match via a tie-break.
However, the Russian was unable to continue after sustaining a leg injury, keeping Djokovic in the hunt for back-to-back titles when he competes in the showpiece, where he will face Stefanos Tsitsipas.
"I really hope that his injury is nothing too serious,” said Djokovic. "I know Daniil, he's a great guy, he's a fighter, he's a big competitor.
"He wouldn't retire a match if he didn't feel that he [could] continue or [if it wouldn't] worsen his injury. He told me that he pulled an adductor muscle in his leg.
"It was such a close match, particularly in the second set. I would probably say he was a better player on the court in both sets. I was fighting and trying to find a way.
"I found a way to win the second, but I'm just sad for the tournament and for these people who were enjoying the battle, and for Daniil that it had to end this way."
Third seed Tsitsipas started slowly in the first set against Andrey Rublev but managed to triumph 4-6 6-4 6-3 in Saturday's other semi-final.
"My mentality kind of changed," said Tsitsipas, who will seek his first ATP 500 title at the ninth attempt. "I played with more of an aggressive game style and didn’t have anything by luck."
At the Japan Open, Taylor Fritz battled back from 3-1 down in the deciding set against Denis Shapovalov to claim a 6-3 6-7 (5-7) 6-3 semi-final victory and book his place in his third ATP Tour final this season.
Third seed Fritz faces an all-American clash in Sunday's final against Frances Tiafoe, who dropped his first set of the week as Soonwoo Kwon bageled him, but the US Open semi-finalist recorded a 6-2 0-6 6-4 win.
"It was a weird match, but I'm happy to get through," said Tiafoe, who progressed into his second final of the season and fifth overall.
"It's not always going to be pretty, it's not always going to be the best stuff, but a win is what matters."
Djokovic complemented a fine display of serving by converting three of his five break points in a 6-4 6-3 victory, recording his seventh consecutive win following his absence from the US Open.
The 21-time grand slam champion has now reached six semi-finals this year, but saw room for improvement in his performance after beating the world number 18.
"I didn't play as well from the baseline as I did in the first two matches, but still it was enough," Djokovic said. "I managed to produce some good tennis when it was most needed in both sets."
Looking ahead to his final-four match, the Serb added: "I always expect highs from myself. Hopefully I can elevate still the level of my game for tomorrow, because it's going to be needed."
Djokovic will face an 11th tour-level meeting with Medvedev in the next round after his fellow former world number one recorded a dominant 6-1 6-1 win over Roberto Bautista Agut.
On the other side of the draw, third seed Stefanos Tsitsipas beat Hubert Hurkacz 7-6 (10-8) 6-3, and will now face Andrey Rublev after he eased to a 6-1 6-2 win over Adrian Mannarino.
At the Japan Open, a knee injury forced Wimbledon runner-up Nick Kyrgios to withdraw from his meeting with Taylor Fritz, with the American set to take on Denis Shapovalov in the semi-finals.
Seventh seed Shapovalov saw off Borna Coric 6-4 6-3 to reach the final four, while fourth seed Frances Tiafoe advanced with a 6-0 6-4 win against Miomir Kecmanovic.
South Korea's Kwon Soon-woo claimed the final place in the semi-finals in Tokyo, beating Spain's Pedro Martinez 6-3 6-0
The 24-year-old beat Tiafoe in straight sets, though it was a tight affair as the match was decided via two tie-breaks, Fritz eventually triumphing 7-6 (7-3) 7-6 (7-2) to clinch the ATP 500 event.
Nevertheless, Fritz looked the composed player for much of the contest as he created eight break chances to Tiafoe's three – though the latter managed to hang in for the most part.
The only breaks of serve came in back-to-back games early in the first set, with Tiafoe quickly hitting back before later holding off two set points to force a tie-break.
He would have been confident going into it after winning 13 successive singles tie-breaks prior to Sunday, but Fritz blew Tiafoe away in both, with the second coming at the end of a set in which neither player gave an inch.
A third tournament victory of 2022 ensures Fritz will break into the top 10 on Monday and provides a real boost to his hopes of qualifying for the end-of-season ATP Finals.
"Crazy, I don't even think it's set in just how fast the last four or five days have been," Fritz said afterwards.
"It's so crazy, and I couldn't have written it any better. It's exactly what I needed for the race [to Turin for the ATP Finals], for my ranking, to kind of put me in a good position for the end of the year, so it's amazing."
Fritz will be eighth in the world when the new rankings are released, making him the first American man to break the top 10 since Jack Sock did so in 2017.
