Chants of 'you're getting sacked in the morning' rang out from the disgruntled away end at the Amex Stadium before a 'Rodgers Out' banner was unfurled at full-time following the Foxes' fifth-successive Premier League loss.
City led with just 51 seconds on the clock thanks to Kelechi Iheanacho and were level at 2-2 at the break but have taken only one point from a possible 18 after being outclassed in the second period.
Rodgers, who was unable to significantly strengthen his squad during the transfer window, says he has not sought reassurances from the club's owners about his future and retains belief in his ability.
"I've never asked for it in all my time here," he replied, when questioned about the backing of the board. "I've never wanted it.
"I will continue to do my work and come in and study and do as much as I possibly can. Until someone tells me differently, I will continue to do that.
"The challenge, we have to embrace it; I'm certainly not going to shirk it. It has been really, really difficult.
"I never lose any belief in what I do. But ultimately, I'm responsible for the results and results at this moment in time haven't been good enough.
"It's a big challenge but it's a great challenge. It's a harsh league if you're not at the top of your game."
Rodgers this week admitted his club's primary objective this term was avoiding relegation following their dismal start and the lack of new signings.
After Iheanacho's rapid opener was overturned by a Luke Thomas own goal and a Moises Caicedo strike, Patson Daka equalised before the break.
But the Foxes were overpowered in the second period and ultimately suffered a heavy defeat following a Leandro Trossard strike and Alexis Mac Allister's set-piece double - a penalty followed by a superb free-kick in added time.
Rodgers urged the club's aggrieved fans to stick behind his underperforming players.
"(It was a) tough afternoon," said the Northern Irishman, who was unable to give a debut to deadline-day signing Wout Faes in Sussex due to a visa issue.
"You can only be honest and Brighton were the better team today.
"It's a difficult moment for the supporters.
"All I encourage is that when it's 0-0 or 1-0 behind, that they keep pushing the players, keep supporting them because it's so important for the players on the field that they feel that support. Otherwise, the anxiety comes in.
"I'm not sure there are too many players that will improve and get better if they don't have that support."
Brighton's emphatic victory tightened their grip on fourth spot.
The rampant Seagulls responded well to falling behind and the frustration of Mac Allister having a thunderous strike ruled out for offside against Enock Mwepu at 2-2 following a lengthy VAR check.
Albion head coach Graham Potter, whose side suffered a first defeat of the season at Fulham in midweek, said: "Great performance, great result, great atmosphere.
"We spoke about responding to the disappointment of midweek and to concede so early is not the ideal start but credit to the boys, they were fantastic.
"We reset at half-time, played really well second half and overall, thoroughly deserved to win."