Kompany has joined Bayern on a three-year deal, and comes in as Thomas Tuchel's replacement.
The former Manchester City captain, who began his coaching career in Belgium with Anderlecht, could not keep Burnley in the Premier League in 2023-24.
Indeed, the Clarets were relegated following a 2-1 defeat to Nottingham Forest at Turf Moor in their penultimate match of the season.
Yet, less than two weeks after the campaign's conclusion, Kompany has been installed as boss of one of Europe's biggest clubs.
The 38-year-old joins Bayern after Tuchel confirmed talks over potentially reversing the decision for him and the club to part ways, which came in February, had ended.
Bayern lost their last game of the Bundesliga season to finish in third place, having failed to win any of the four trophies they were in contention for, seeing their dominance of Germany's top tier ended by Xabi Alonso's unbeaten Bayer Leverkusen team.
Alonso had been Bayern's primary target, but their former midfielder has elected to stay at Leverkusen for next season.
Bayern were also linked with Roberto De Zerbi, who has just left Brighton, and Brentford coach Thomas Frank, while they were turned down by Austria coach Ralf Rangnick. Mauricio Pochettino, who left Chelsea on Monday, was also rumoured as a potential option.
Kompany, then, is a surprise pick, but Bayern will be hoping he rediscovers the magic touch he had during his first season at Turf Moor in 2022-23, when he guided the Clarets to promotion from the Championship, earning 101 points in the process.
However, Kompany's possession-based, front-foot approach was badly exposed in the Premier League, with Burnley winning just five games on their way to taking 24 points as they finished 19th, going back down alongside Luton Town and Sheffield United.
He leaves Burnley having managed 96 games, winning 41 (42.7 per cent) of those matches.
In a statement, Burnley said they were "initially confident" of keeping Kompany at the club, but claimed "the changing dynamics of the situation made this impossible".
"We understand the allure and prestige of a club like Bayern Munich and respect Vincent's ambition to explore new opportunities," the statement continued.
"We wish only the best for Vincent and would like to put on record our appreciation for his dedication every single day he was a part of this football club.
"Our priority remains the stability and success of Burnley, and we will continue to take the necessary steps to ensure that our ambition is realised, namely our return to the Premier League."
Bellamy, who talked extensively about his long battle with depression last year, had been part of the Anderlecht coaching staff since 2019.
The 42-year-old has been working as former Manchester City team-mate Kompany's right-hand man since earlier this year, but he will officially step aside on Monday.
A statement on Anderlecht's website on Sunday read: "Assistant coach Craig Bellamy will leave the coaching staff of RSC Anderlecht on Monday.
"Bellamy already struggled with mental problems in the past and these have resurfaced again recently.
"With his departure, the Brit took a very difficult, but absolutely necessary decision concerning his health.
"Bellamy said goodbye to the players in the dressing room on Sunday afternoon after the 7-2 victory against KV Mechelen."
Former Wales international Bellamy retired from football in 2014 following a second spell with hometown club Cardiff City.
He moved into coaching with Cardiff at academy level before departing in January 2019 and taking up a similar position with Anderlecht.
Following the news of Bellamy's departure, Kompany added: "Craig has long struggled with his mental health. Now the monster is back there.
"Health is above football. We are obligated to give him the chance to get well."
The Pro League in early April suggested the 2019-20 season be cancelled and a meeting to determine its fate is due to be held on Friday.
Anderlecht last played on March 7, defeating Zulte-Waregem 7-0 at home, though Nasri did not feature in the matchday squad.
Nasri has been at the centre of media rumours this week, however, after reports in Belgian publication Derniere Heure claimed the former Manchester City and Arsenal playmaker had not been in regular contact with player-manager Vincent Kompany.
But Nasri has dismissed the rumours, insisting he has been keeping in touch with Anderlecht.
"It makes no sense," Nasri said during in a live interview on Instagram.
"Where did you see someone disappear? Because that is what I thought I saw or heard said: someone who has disappeared or no longer gives any sign of life.
"I speak to the chief medical officer every day. He is the one who sends me the sessions I have to do. I also speak with Vincent Kompany."
Nasri joined up with former City team-mate Kompany at Anderlecht in 2019 but has failed to revive his career in Belgium, having made just eight appearances in all competitions.
Player-boss Kompany has seen his players stay fit at home in recent weeks, with the Anderlecht stars dialling in on Zoom to take part in exercise sessions.
But many first-team footballers are expected back at the Neerpede training ground after Anderlecht decided the time was right to reunite in the same place.
Belgium remains largely locked down due to the coronavirus pandemic, and it was announced on April 2 that a plan was in place to abandon the Pro League season; however, a final decision in that respect has yet to be taken. There is only one round of regular-season games remaining.
In the meantime, Anderlecht have invited players to pull on their training kit and assemble together, while stressing attendance on Monday is voluntary.
Tackling and physical contact is set to be banned in the sessions.
Club spokesman David Steegen said: "Players who do not feel reassured can continue to work at home. But each player present at Neerpede will observe the safety distances and the coaches will direct the exercises while respecting the social distance."
Speaking to Belgian public service broadcaster RTBF, Steegen added: "It will be mainly physical training, but also with a ball.
"All the equipment used will be disinfected, before and after each session. Players will not have access to the dressing room, everyone will change and shower at home."
UEFA questioned Belgium's intention to cancel the rest of its league season, and there will be no final ruling on whether the campaign will resume until late April at the earliest.
Anderlecht have endured a rocky first season since Kompany joined after bringing an end to his illustrious Manchester City career.
They sit eighth in the Pro League, with Club Brugge 15 points clear of closest rivals Gent at the top.
The Pro League in early April suggested the 2019-20 season be cancelled and a meeting to determine its fate is due to be held on Friday.
Anderlecht last played on March 7, defeating Zulte-Waregem 7-0 at home, though Nasri did not feature in the matchday squad.
Nasri has been at the centre of media rumours this week, however, after reports in Belgian publication Derniere Heure claimed the former Manchester City and Arsenal playmaker had not been in regular contact with player-manager Vincent Kompany.
But Nasri has dismissed the rumours, insisting he has been keeping in touch with Anderlecht.
"It makes no sense," Nasri said during in a live interview on Instagram.
"Where did you see someone disappear? Because that is what I thought I saw or heard said: someone who has disappeared or no longer gives any sign of life.
"I speak to the chief medical officer every day. He is the one who sends me the sessions I have to do. I also speak with Vincent Kompany."
Nasri joined up with former City team-mate Kompany at Anderlecht in 2019 but has failed to revive his career in Belgium, having made just eight appearances in all competitions.
The former Manchester City captain, now manager of Anderlecht, stated after the game that players and coaches were verbally abused throughout.
"I go home disgusted and disappointed. My players, my staff and I were victims of racist insults," the 35-year-old told broadcaster Eleven Sports.
"I want to get together with my staff, to be with the people who matter to me. We should not still have to experience this today."
Club Brugge, who were in the same Champions League group as Manchester City this season, released a statement after the game condemning the actions of their fans, saying: "Club Brugge, its fans, staff, players and board, strongly condemn any form of racism.
"These individuals are not representative of the values and norms of our club, and do not have their place at Jan Breydel Stadium."
On Monday, Club Brugge said the club would do all they can to identify those responsible and seek to impose stadium bans.
Kompany won two Belgian league titles as a player at Anderlecht before going on to win four Premier League titles with City, as well as two FA Cups and four EFL Cups during his time at the Etihad Stadium, before heading back to Anderlecht as player-coach in 2019. He retired from playing in 2020 to focus on managerial duties.
Chelsea striker Romelu Lukaku posted support for his former Belgium team-mate on Instagram, and demanded a firm response from the football authorities.
"An icon like Vincent Kompany has been insulted because of his skin colour," Lukaku wrote. "Enough is enough... take real action now."F
Pep Guardiola's top-flight champions eased past Bristol City 3-0 on Tuesday, while United backed up their EFL Cup success with a 3-1 comeback victory over West Ham.
City will face former defender Kompany, who lifted four Premier League titles at Etihad Stadium, after being drawn at home to runaway Championship leaders Burnley following their late 1-0 win against Fleetwood Town on Wednesday.
United ended a six-year trophy drought on Sunday by defeating Newcastle United and will continue their quest for further silverware when they host Fulham at Old Trafford.
Fourth-tier Grimsby Town remain the shock package in this season's FA Cup after Premier League Southampton fell to a 2-1 home loss and became the Mariners' fifth victim from a higher division in the 2022-23 campaign.
The League Two side's reward will be a trip to in-form Brighton and Hove Albion, who edged past Stoke City 1-0 in the fifth round on Tuesday.
Sheffield United were another surprise winner in the last 16 of the competition, defeating Tottenham 1-0, and will host Blackburn Rovers, which ensures at least one Championship side will reach the semi-finals.
All ties are set to be played between March 17 and 19 with the quarter-final contestants one win from a trip to Wembley.
FA Cup quarter-final draw in full:
Manchester City v Burnley
Manchester United v Fulham
Brighton and Hove Albion v Grimsby Town
Sheffield United v Blackburn Rovers