With several questions asked of their defence, having conceded in all but one of their preceding 12 league games, the Hammers held their own at the London Stadium as both sides shared the spoils.
The point leaves Slaven Bilic's men on the 14th position, four places above the relegation zone but eight spots below the Toffees.
“We’ve had a lot of injuries so first and foremost, we had to stop them scoring,” Reid told his club's official website.
“Then, going forward, we had to get Manu [Lanzini] and Dre [Ayew], those types of players on the ball and try to get into good positions.
“I guess that final ball wasn’t always there and the sharpness, but these games earlier in the season we would have lost and we’re pleased to have come away with the point after working hard, which is what you always want.
“It was a decent performance from the team. As long as we keep up this good work then we can get the points that we need from now until the end of the season.”
Everton had found the back of the net in each of their preceding eight matches.
Ayew, who joined West Ham on a £20.5 million club record deal, has so far made 21 topflight outings with 12 starts for the Hammers.
The 27-year-old has five goals to his credit, with four of the five coming in his last nine matches.