Sean Dyche's in-form troops were on course for another three-point haul when Vitaliy Mykolenko struck early doors, only for an unfortunate Ashley Young own goal to see the Seagulls take a slice of the spoils home to the South Coast.
Eleven goals flew into the back of the net during the sides' two meetings last term, where Brighton's 4-1 Goodison Park thumping preceded Everton's astonishing 5-1 Amex annihilation, and the early exchanges hinted that another goal-laden affair could be in the offing.
With seven minutes gone, Dwight McNeil kept the ball alive on the left and sent in a deflected cross to Mykolenko, who had one strike at the near post kept out by Bart Verbruggen, but the Ukrainian followed up and beat the Dutchman at the second attempt with the help of a deflection off of Lewis Dunk.
The Brighton defender appeared to have made amends in the 15th minute with a sumptuous left-footed volley from a Pascal Gross free kick, but following a VAR check, Dunk was denied his slick leveller by a marginal offside.
A galvanised Seagulls continued to come forward in droves, and the feet of Pickford were called into action to deny Kaoru Mitoma in the 28th minute, but Roberto De Zerbi's side were still lacking that final ball to properly hurt Everton.
The Toffees appealed for a spot kick in the 42nd minute when Joel Veltman barged McNeil in the back right on the line, but referee Tim Robinson waved away the protests, and De Zerbi brought Joao Pedro on at the break in an attempt to inject some attacking bite into his troops.
The visitors continued to enjoy the lion's share of the ball at Goodison and threatened through Dunk once again in the 56th minute, but Pickford was equal to his England teammate's looping effort.
The Three Lions number one was otherwise offered staunch protection from his backline, and Brighton were soon on the ropes again, as McNeil fired wide in the 76th minute before Dunk denied Dominic Calvert-Lewin a simple finish from Abdoulaye Doucoure's cross one moment later.
However, Brighton ensured that their perfect scoring streak in the 2023-24 season remained alive in the 84th minute, when Mitoma attempted to pick out a teammate with a cross from the left but saw his delivery strike Young and sail into the Everton net over Pickford.
Everton had the better chances heading towards injury time, but the two coaches shook hands on a point, which sees the Toffees remain in 15th place and Brighton rise into sixth spot above Newcastle United.
A trip to Crystal Palace is on the menu for Everton in seven days' time, while Brighton continue their Europa League campaign away to a below-par Ajax on Thursday evening.