Sesko double not enough as Man Utd stumble at Burnley
Ayden Heaven's own goal and Jaidon Anthony's fine strike secured a point for second-bottom Burnley, who remain without a win since late October.
United blew another chance to climb into the Premier League's top four after failing to win for the fifth time in six games.
That series of disappointing results combined with a clash with the club's director of football Jason Wilcox saw United part company with Amorim on Monday.
Former United midfielder Darren Fletcher took caretaker charge, with his former team-mates Ole Gunnar Solskjaer and Michael Carrick reportedly among the leading candidates to be appointed interim manager until the end of the season.
Fletcher made an immediate change to a back four as the 3-4-3 system that proved Amorim's downfall was abandoned.
The visitors were also helped by the return of captain Bruno Fernandes from injury, while Fletcher named his son Jack among the substitutes.
Burnley have not won in 12 games to fall eight points adrift of safety, but also ran United close when the sides met at Old Trafford earlier in the season when a stoppage time Fernandes penalty secured a 3-2 win for the Red Devils.
The home side got a big slice of fortune to take the lead when Bashir Humphreys' cross deflected off Heaven and looped into the far corner.
United's bad luck continued when Lisandro Martinez had an equaliser controversially ruled out for a foul on Kyle Walker.
Humphries and Maxime Esteve also made brilliant goal-line clearances to preserve Burnley's lead until half-time.
But Sesko took just 10 minutes to double his tally of goals since joining United for up to £74 million ($100 million) from RB Leipzig from two to four.
The Slovenian striker lashed home from Fernandes' clever through ball before expertly diverting Patrick Dorgu's cross beyond Martin Dubravka.
Fernandes was instantly withdrawn by Fletcher to protect the Portuguese on his return from a hamstring injury.
But his absence was felt in the final half-hour as United threw away another two points in the battle for a place in next season's Champions League.
Anthony was allowed too much space to turn before firing brilliantly into the top corner to level.
United still came close to snatching all three points when teenager Shea Lacey crashed a long range strike off the crossbar.
But they dropped to seventh, slipping behind Brentford and Newcastle, and are two points off fourth-placed Liverpool.