When Brighton's fans sang "Your city is blue" in the final moments of the game, humiliation for United and Mourinho was complete.
Before this game Mourinho accused Manchester City of a lack of class after learning that a documentary about the Premier League champions featured digs at his expense. Bailly, not to be outdone, gave away a corner in the buildup to the second and then chopped down Gross in the box for Brighton's penalty just before half-time.
United lie in 10th place after the first two rounds of Premier League action, having accrued three points so far.
United improved marginally after the break but never really looked like threatening a fightback, though Paul Pogba netted an injury-time penalty after Duffy brought down Marouane Fellaini.
Both signed by Mourinho for a combined fee of £60 million, they have started the season as first choice, but it would be no surprise if they were both sat in the stands when Tottenham turn up at Old Trafford on August 27.
"We were punished by the mistakes we made".
"We were playing at home last week and that's different".
First half goals from Glenn Murray, Shane Duffy and Pascal Gros left United in a state of disarray and despite a 34th minute strike from Romelu Lukaku, they failed to cause any problems for large spells.
This confirmation of captaincy puts to bed speculation that United could install Pogba as the first-choice captain this season after his debut in the armband last weekend.
Bailly upended Gross from behind to concede a penalty that the Austrian converted past David De Gea.
'Another goal could have changed things again but our second goal arrived too late.
Mourinho will now demand an instant reaction when United hosts Tottenham in its next game.
"He looked fit. He looked leaner".
Former Burnley forward Andre Gray opened the scoring in the third minute with a superbly taken volley. The Brazilian midfielder was at his marauding best at times and will be looking to continue his solid start for the Red Devils.
Burnley have now gone seven league games without a win.