WATCH Everton and Crystal Palace in the 26/27 Carabao Cup, & FA Cup in NZ, LIVE & EXCLUSIVE on beIN SPORTS | Sign up here and save up to $60 on our Annual Pass Now | AU | NZ

Dewsbury-Hall found the top-right corner with a thunderous strike shortly before half-time, with Everton punishing some wasteful finishing from its visitors after Eddie Nketiah and Jean-Philippe Mateta both hit the same post.

Barry got the clinching goal early in the second period, with the outstanding Iliman Ndiaye giving his striker a simple headed finish with a pinpoint cross.

The game started in a wide-open fashion, and it was Palace that carved out the first two opportunities as Mateta flicked Nketiah's centre over and Daniel Munoz saw a left-footed piledriver pushed away by Jordan Pickford.

Everton's first big chance saw Ndiaye work Dean Henderson, but the Toffees were fortunate not to fall behind when Pickford brilliantly tipped Nketiah's shot on to his left-hand post, after some incisive work from Adam Wharton.

The hosts had another lucky escape when Mateta side-footed against the same post in the 38th minute, and they took full advantage only four minutes later, with Dewsbury-Hall rifling across goal and in after some excellent out-of-possession work from Harrison Armstrong.

Everton's second goal came eight minutes into the second period as Barry nodded home after being picked out by Ndiaye, who could have put the result beyond all doubt soon afterwards, when his volley skipped off the turf and was touched over by Henderson.

The closest Palace came to a consolation goal was when substitute Jorgen Strand Larsen headed straight at Pickford in stoppage-time, and a disappointing start to its season was capped by Chadi Riad being carried off on a stretcher with what appeared to be a serious knee injury.

Everton was arguably fortunate to stay in the game prior to Dewsbury-Hall's rasping 42nd-minute strike, and Toffees boss David Moyes knew it.

"Both teams had periods in the game. We could easily have been a couple of goals down – it was a worldie save and it could have gone the other way," Moyes said.

"Once we got the first goal, it settled us. That mattered. We had big moments; Kiernan's goal was important.

"We managed the game well after that and contained Palace really well."

Palace's wastefulness was reflected in the fact that they won the expected goals battle, even while having five fewer shots than Everton, and new boss Sage lamented a drop in standards in the second half.

"We played in the right way to win in the first half, but not in the second," Sage said. "But when you play in this direction, you have to score.

"We missed two or three big opportunities, and we gave the goal to the opponent. That's why the second half was difficult, with them getting the second goal too.

"Sometimes, you know, you do the job but you don't get the salary at the end – and today it was like that.

"Sometimes you play very well and you can keep it on YouTube, and when you lose your finishing, you take the clip and you put it in the waste. That's why a goal is very important."