The Toffees went into the contest as one of three teams who could still fill the two remaining relegation places, but a stunning strike from Abdoulaye Doucoure got them the win they needed to preserve a top-flight status that has now lasted 70 years.
It was not plain-sailing for Sean Dyche's men, though, and at one stage they dropped into the bottom three while their match was still goalless and Leicester City had taken the lead against West Ham United.
Despite a tense atmosphere around Goodison Park, Doucoure's thunderous and momentous strike sent the fans into raptures and ultimately saw Leicester and Leeds United suffer relegation from the top flight, as Everton survived on the final day for the third time in the Premier League era.
Everton went into the contest with their fate in their own hands and knowing that victory would be guaranteed to prevent them from dropping into the second tier for the first time in 70 years.
Goodison Park was in fine voice to roar their players towards that goal, but the hosts struggled to create clear chances in a first half that became more tense as it went on.
News of Leicester taking the lead against West Ham dampened the atmosphere at Goodison Park, with Everton dropping into the relegation zone and suddenly needing a goal of their own to stay up.
James Garner almost provided it at the end of the first half when he sent a dipping, curling effort towards the top corner, but once again Travers got his fingertips to it to keep it out.
The feeling from Everton that it may not have been their day only grew in the 51st minute when Demarai Gray squandered a clear chance, not getting enough on his header as Travers again made a crucial save.
However, the all-important moment arrived just before the hour mark, with Doucoure securing himself hero status by hammering a cleanly-struck half-volley past the unsighted keeper to crank the volume back up to max inside Goodison Park.
While that memorable strike lifted the Toffees back out of the relegation zone, their position was still perilous given Leicester's lead over West Ham, with one Bournemouth goal sending them back into the bottom three.
As a result, 10 minutes of stoppage time was signalled, and there was one final scare for Everton when Matias Vina's powerful shot was beaten away by Pickford four minutes into that additional period.
However, the Toffees safely held on for the crucial three points which see them finish in 17th, two points clear of Leicester, whose victory over West Ham was ultimately in vain.No Data Analysis info