Elliot started the Premier League game at St. James’ Park and played for the entire 90 minutes, as Newcastle suffered a 1-0 defeat.
The Republic of Ireland international goalkeeper made as many as eight saves over the course of the 90 minutes.
"Newcastle feel like they have been mugged, they had the better opportunities in the second half,” said Lawrenson on BBC Radio 5 live. “Elliot has been outstanding in the second half, but his punch wasn't too clever.
"This should have been a draw, Newcastle are really unlucky but well done to Everton, who stuck at it and won the game.”
Over the course of the 90 minutes at St. James’ Park, hosts Newcastle had 41% of the possession, took 10 shots of which just one was on target, and earned three corners.
Visitors Everton had 59% of the possession, took 20 shots of which just seven were on target, and earned seven corners.
Newcastle are third from bottom in the Premier League table at the moment with 17 points from 18 matches, level on points with 17th-placed Norwich City.
Steve McClaren's side will return to action on Monday afternoon when they take on West Bromwich Albion away from home at The Hawthorns in the league.