Gary O'Neil's side looked bright to start with, but fell behind when Eze fired his side ahead late in the first half.
It was not all good news for Roy Hodgson's hosts, who saw Wilfried Zaha forced off after the break. However, Eze added a second for the Eagles, who had already secured mathematical safety, while the visiting Cherries still need one more point to mathematically avoid the drop.
Neither side was able to create any clear-cut chances inside the first 10 minutes though Bournemouth frequently tested the fortitude of centre-back pairing Marc Guehi and Joachim Andersen.
Michael Olise curled an effort wide of the far post from the edge of the area before his side were able to stave off a sustained wave of pressure from the opposition as the period ticked past the halfway point. Joel Ward evaded two black shirts in the corner to set up Olise for a second attempt, coming closer with a cross to force Neto into a leaping grab.
Olise entered the afternoon with nine assists and nearly added another when he sent a cross to Wilfried Zaha at the far post, who misdirected his effort from three yards out.
VAR was called in for the first time following an altercation between Andersen and Jefferson Lerma inside the Cherries' area for a potential red card on the Palace defender, but the hosts remained at full strength after the review.
Palace had failed to make anything of six corners to Bournemouth's two, but the statistics were otherwise closely split as the contest approached the 40-minute mark as both sides searched for a breakthrough.
It was the hosts who finally achieved it, sparked by Zaha, who battled his way around his marker at the edge of the area before cutting back to Jordan Ayew. He took a quick backwards touch to direct the ball to Eze, who gave Neto no chance as he fired home from centre after 39 minutes.
Will Hughes quickly tested Neto to start the second half but the Selhurst faithful soon witnessed a worrying scene when Zaha, only recently recovered from a hamstring injury, pulled up and was replaced by Odsonne Edouard. The Eagles talisman, out of contract at the end of this campaign, was treated to a rousing chant by the home support, possibly wondering if this would be the last time they would watch him walk off the pitch in a Palace shirt.
Eze bagged his second in the 58th minute, a solo effort that saw him collect the ball from Olise on the left before a mazy run through centre put him in position to fire his 10th of the season into the top right.
Palace's corner tally ticked over into double-digits but the scoreline stood as the hosts survived a late free-kick from a dangerous position.
Palace survived before substitute Jairo Riedewald tried to make up for his involvement in the set-piece by attempting to tee up Edouard for a late third - but two was ultimately all it took.