Palace threw away a 2-0 half-time lead in the second half, and the dramatic victory for the Toffees ensured its Premier League status for another season.
The delirious home fans descended onto the pitch after the final whistle to celebrate, but one fan approached Vieira and appeared to taunt him.
The former Arsenal midfielder initially ignored the man, before turning to confront him, seeming to kick out before being ushered away by other Everton supporters.
Vieira refused to speak about the incident in his post-match media conference, though his opposite number Frank Lampard said he "felt" for the Frenchman being put in that situation.
"I feel for Patrick," the Everton boss said at his own media conference. "I didn't get him at the end. I would have said, 'come in with us' – although he might not have wanted that.
"Of course he ran 80 yards across the pitch and it's not easy."
This was not the first unsavoury incident to take place in English football in the last week involving pitch invasions, with a Nottingham Forest fan being arrested and later jailed for assaulting Sheffield United striker Billy Sharp after Forest's penalty shoot-out win in the Championship playoff semi-finals.
There were further concerns raised about fans endangering players during pitch invasions after both League Two playoff games involving Mansfield Town and Northampton Town, and Swindon Town and Port Vale.
On the fans coming onto the pitch, Lampard added: "It was pure elation of fans who have come on the pitch after avoiding relegation.
"If it is done in the right way let them stay on the pitch, let them have their moment. As long as they behave, no problem."