Palhinha scored the only goal just before half-time at the Tottenham Hotspur Stadium, scrambling home on the rebound after his header from Mathys Tel's corner struck the post.
Everton pressed throughout the second half and came close to equalising deep in stoppage time, when Tyrique George's curling effort was tipped over by goalkeeper Antonin Kinsky in what proved the decisive moment. West Ham United, who beat Leeds United 3-0 at the London Stadium, were relegated as Spurs held on.
Van de Ven, who captained the side in the absence of the injured Cristian Romero, was in no mood for celebration.
"It is unacceptable that the last game we played this season we play for relegation. This club has some unbelievable players. It was embarrassing to let it come to the final day, but we did it and that is what is important."
He reflected on a season that has cost him and the squad significantly: "To be honest I was more emotional at the end of the final whistle. It has been a tough season for me personally as well. I played almost every game and I have suffered a lot. The emotions are really happy and we must not let it happen again."
Palhinha was more measured but equally aware of the weight of the occasion: "It is an amazing feeling after a really tough season. A lot of things happened in every single way, but today with the crowd I have never seen anything like that. We cannot change the bad season we've had, but this environment and support from the fans is amazing."
He dedicated his goal specifically to his family.
"I had a lot of emotions inside and I dedicate the goal to both my sons. The goal is for them and my parents."
On the future: "The club will grow up after this season. We know what we have to do on the pitch. This victory is for the supporters — they deserved it."
Spurs burned through three managers this season — Thomas Frank, Igor Tudor and Roberto De Zerbi — before Roberto De Zerbi's arrival in late March produced 11 points from seven games. They finish 17th, two points above the relegated Hammers.