Brighton dominated the clash for long spells, but Jarrad Branthwaite's excellent finish in the 73rd minute of the contest looked to be securing a vital three points for Everton in their battle against relegation.
Dunk had other ideas, though, heading home in stoppage time to ensure that the Seagulls would secure a share of the spoils.
Sean Dyche's side remain 17th in the table, one point clear of 18th-placed Luton Town, while Brighton remain seventh, two points ahead of eighth-placed Newcastle United.
Brighton dominated the early exchanges and had a half-chance in the ninth minute through Danny Welbeck after smart work from Tariq Lamptey, but the forward's effort was deflected narrowly wide of the post.
The Seagulls had another chance in the 13th minute, with Welbeck beating Everton goalkeeper Jordan Pickford to a loose ball, but the ex-Manchester United youngster failed to make the most of the opportunity.
Brighton were in control, with Everton offering nothing in the final third, but the Toffees were defending well, with James Tarkowski at the heart of it.
Welbeck had another chance in the 20th minute after a one-two with Facundo Buonanotte, but a last-gasp challenge from Ben Godfrey prevented him from getting a clean strike away.
Simon Adingra then came close six minutes later, with the attacker's left-footed strike just missing the target, before excellent defending from Dunk prevented Dominic Calvert-Lewin from making contact on a dangerous Dwight McNeil cross down the other end.
Despite Brighton's domination, the scores were level at the interval, with Everton performing impressively at the back in the opening 45 minutes of action.
Everton came close to making the breakthrough just before the hour when Abdoulaye Doucoure's shot headed towards the top corner, but Lamptey was on hand to make an excellent headed clearance.
The Toffees took the lead in the 74th minute of the contest through Branthwaite, with the centre-back picking out the top corner with a brilliant finish from inside the Brighton box.
Brighton were reduced to 10 men in the 81st minute, when Billy Gilmour was penalised for a foul on Amadou Onana.
However, the Seagulls found a leveller in the 95th minute through Dunk, who climbed highest inside the penalty box to head a cross from Pascal Gross into the back of the net.
Everton will continue their league campaign at home to West Ham United next Saturday, while Brighton's attention will now switch to the FA Cup, preparing to head to Wolverhampton Wanderers in the fifth round of the competition on Wednesday night.