Making amends for their harrowing midweek loss to Luton Town, Brighton ran riot in the first half and entered the break three goals to the good thanks to Lewis Dunk, Jack Hinshelwood and Facundo Buonanotte.
Palace's afternoon went from bad to worse when Michael Olise went off injured just nine minutes after being introduced, and even though Jean-Philippe Mateta briefly reduced the deficit, Joao Pedro restored the hosts' three-goal advantage and potentially hammered another nail into Hodgson's Eagles coffin.
Following their enforced withdrawals in Palace's beating of Sheffield United, Olise was only fit enough for a place on the Eagles bench alongside new signing Adam Wharton, but there was no Eberechi Eze in sight.
Four changes on the Brighton end was hardly a surprise in the wake of their Luton embarrassment, where they fell behind with just one minute on the clock, but only three minutes were on the board at the Amex by the time that the Seagulls broke the deadlock.
In front of a hostile away end, Pascal Gross delivered an inswinging corner towards the head of Dunk, who comfortably towered over Joachim Andersen and left Dean Henderson rooted to the spot with a simple header into the ground and into the net.
Timed at two minutes and 22 seconds, Dunk's header represented Brighton's quickest goal of the current top-flight campaign, and Roberto De Zerbi's dominant men did not rest on their laurels as the half wore on.
Palace's injury crisis worsened in the 27th minute when Marc Guehi was forced off with a knee problem - Chris Richards took his place in defence as Wharton slotted into midfield for his debut - and a rampant Brighton quickly took advantage of the Eagles' predicament.
With 32 minutes on the clock, Wharton charged down a Gross strike, but the ball fell kindly for Tariq Lamptey, who sent in an inviting delivery from the left, and fellow full-back Hinshelwood connected among a sea of bodies and directed his header into the far side of the net.
There would be no respite for a beleaguered Palace, as just 83 seconds after Hinshelwood doubled Brighton's lead, the Seagulls had their third, as Gross robbed Wharton of the ball and burst forward into the box before teeing up Buonanotte, whose first-time curler sailed into the back of the net.
The despondent away end exhibited a banner emblazoned with the words "No shared vision. No structured plan. Parish out." as Brighton - who had evidently taken a leaf from Luton's book - blew their side to smithereens in the first half, convincing Hodgson to bite the bullet and bring on Olise for Jeffrey Schlupp at the break.
However, the Palace boss' questionable decision backfired horribly, as just nine minutes after the restart, Olise pulled up with an apparent hamstring issue and was forced to call it a day, trudging down the tunnel to be replaced by Matheus Franca.
Wild challenges began to fly in left, right and centre from the exasperated visitors, but there would be a silver lining in the 72nd minute, as Mateta met Andersen's cross with a venomous near-post header to set up an intriguing finale.
Any late nerves in the home end would be extinguished in the 85th minute, though, as Pedro and Danny Welbeck played a smart one-two on the edge of the Palace box, with the latter backheeling the ball into the Brazilian's path before Pedro finished with aplomb.
The Eagles' defeat has no effect on their 14th-placed ranking in the Premier League table, although they are only five points clear of the drop zone, while De Zerbi's men shoot above Manchester United and Newcastle United into seventh spot thanks to the Magpies' absorbing 4-4 draw with Luton Town.
Next weekend, Brighton hit the road once again for a date with Tottenham Hotspur, two days before Palace host Chelsea at Selhurst Park on February 12, although only time will tell whether Hodgson remains in control of the reins for that London derby.