The game pitched together two former Netherlands team-mates with Frank de Boer in charge of Atlanta - and it was Stam who prevailed in the head-to-head contest with his compatriot thanks to a 78th-minute strike.
Amaya's goal was worthy of settling the contest too, as he curled home with his left foot from just outside the penalty area.
Atlanta - who have now suffered back-to-back defeats in Group E - were already a man light at that stage, Jake Mulraney receiving two yellow cards inside the opening 26 minutes at the ESPN Wide World of Sports Complex.
De Boer then saw substitute J.J. Williams dismissed deep into 13 minutes of additional time for kicking out at Kendall Waston.
The former Manchester United, Milan and Netherlands defender heads to Ohio along with Said Bakkati, who has worked as Stam's assistant throughout his coaching career.
"For myself, to get the opportunity to work in America and to work in MLS for a club like FC Cincinnati, it's a dream," Stam told his new club's website.
"I've always liked the country and I've always liked the mentality of the people in the States as well in what they're trying to achieve.
"To get the opportunity to work for FC Cincinnati and this project by building the club – hopefully even further than where they already are, because I think they've already done a great job – but being a presence as a team, even more in MLS, getting results and working together with the fans and everybody already working within the club, for me it's a big challenge to do well."
Stam started his post-playing career with Jong Ajax in 2014 before spending two years in charge of English Championship side Reading.
He joined PEC Zwolle for 2018-19 before taking over at Feyenoord in June 2019, only to step down three months later after a 4-0 loss to Ajax.
"We are very happy and pleased to have Jaap Stam become FC Cincinnati's new head coach," said general manager Gerard Nijkamp.
"He has an extensive background in being part of some of the most successful clubs in the world as a player and has also proven that he can guide teams through transitionary periods as a head coach.
"I strongly believe that Jaap can lead us through our next phase, to our new stadium, and help us achieve our goals in the future, making players better and bringing his knowledge and experience inside the club. I have full confidence that he will be successful at FC Cincinnati and will be a great fit to carry out our established playing philosophy and guide the club to achieve future successes."
Cincinnati finished bottom of the Eastern Conference with six wins from 34 games in 2019, their first year in MLS following the awarding of an expansion franchise to the city in 2018.
Announcing Stam as their boss on Thursday, Cincinnati revealed the news with a picture of a man who certainly looked rather like the former Manchester United and Netherlands defender.
The man in question looked dressed for the job, in shirt, tie and a large coat, and as well as a bald head, many of his facial characteristics were comparable to those of Stam.
However, it turned out to be not Stam but a man widely identified by Dutch media as Ajax youth coach Tinus van Teunenbroek.
Red-faced Cincinnati pulled Van Teunenbroek's image from their coverage of the story once it became clear they had the wrong man.
A corrected version of the announcement was posted to Twitter, this time with an image of Stam, accompanied by the message: "Join us in welcoming our actual new head coach, Jaap Stam."
The former Manchester United and Netherlands defender departs with Cincinnati second bottom in the Eastern Conference after just four wins from 25 games.
Tyrone Marshall, who is head coach of the Under-19 team, takes over on an interim basis while a permanent replacement is sought.
Stam joined the club in May 2020 and presided over 47 games with the MLS club, but won just eight, drawing 13 and losing 26.
The team's poor form forced club president Jeff Berding to take action, with Stam followed out of the door by assistant coaches Said Bakkati and Yoann Damet.
"During our nearly two-month process to identify the next general manager for FC Cincinnati, it was apparent that a head coach change was necessary. Therefore, after a great deal of deliberation, the club has made the decision to now move to the future with Jaap no longer serving as head coach," Berding said.
"We believe a change in leadership is in the best interest of the club at this time, and a new general manager will lead the search for a new head coach. We thank Jaap for everything he has done for FC Cincinnati during his time here, and we wish him the best as he can now head home to be with his family."
Berding also had words of support for former Jamaica international Marshall, who joined the club this year from Real Salt Lake.
"Tyrone has been a valuable addition to our coaching staff since he arrived earlier this year with plans to be the future coach of our MLS2 team," said Berding.
"He has shown his coaching ability and leadership qualities, and he is the right person to inject new energy into the locker room and help our squad finish 2021 strong with an eye to the future."
Jaap Stam's Cincinnati bounced back from their opening-game loss in Group E with back-to-back wins, including Wednesday's 2-0 victory over New York Red Bulls at the Walt Disney World Resort in Florida.
Yuya Kubo opened the scoring just before half-time with a good finish from the edge of the area after poor defending by the Red Bulls.
Florian Valot's own goal early in the second half sealed Cincinnati's win and second place in Group E.
The result also meant Thierry Henry's Montreal Impact advanced as one of the best third-placed teams.
Already into the last 16 from Group D, Minnesota were unable to claim top spot after a 2-2 draw with Colorado Rapids.
Kei Kamara opened the scoring for the Rapids, but they went into half-time behind.
Ethan Finlay flicked in the equaliser for Minnesota before tapping in a second.
Jonathan Lewis' powerful strike brought Colorado level as they secured their first and only point of the tournament.
Cincinnati announced Stam as their new coach on Thursday, with their Twitter post accompanied by a composite image that included a man who resembled the Dutchman.
But the pictured man, who was also bald and dressed in a shirt, tie and large coat, turned out to be Ajax youth coach Tinus van Teunenbroek.
The mishap quickly went viral and left Cincinnati's media team red-faced, particularly after official accounts of fellow MLS clubs chimed in on the replies of the corrected follow-up tweet.
Each one posted a welcome message and an accompanying image of a different bald man – examples ranged from Jason Statham to Dr. Evil from the Austin Powers films.
Unsurprisingly Stam was asked about the mishap during his presentation news conference on Friday, but the former Manchester United and Milan centre-back, who cut an intimidating figure on the pitch, took it in good humour.
"Everything was going well and I know you're aiming a little bit at this picture that came out," Stam said when asked how it felt to 'go viral'.
"Yesterday [Thursday] was a busy day, a hard day, a lot of talking and a lot of meetings, so eventually we used a doppelganger to do the picture."
Stam was then asked what he thought about the MLS clubs' replies. He said: "I've seen it – somebody sent it to me and I can appreciate that as well.
"At a certain time we need to have a bit of a laugh. If we play against each other and we come out on top with a win, then we can have a laugh – that's what we're trying to do."
The Dutch journalist who asked the questions explained to Cincinnati's press officer that he was addressing the "elephant in the room", but further questions on the matter seemed to be discretely discouraged when she responded: "Well, no, we have Jaap in the room today, so we're good there."
Stam is returning to management for the first time since resigning as Feyenoord boss in October, as he quit following a 4-0 defeat by Ajax less than five months after succeeding Giovanni van Bronckhorst.
Cincinnati are in their second MLS season and lost the first two matches of the 2020 Eastern Conference campaign prior to the coronavirus-enforced pause.
