But Newcastle had a golden chance to level just before the half hour mark when Everton goalkeeper Jordan Pickford brought down Salomon Rondon, having first spilled a cross from Matt Ritche and tried to recover it.
"What an fantastic feeling", the Spanish attacker said.
"There are still a few games to go but we are closer [to safety]". If we did what we should've in the first 15 minutes of the second-half will we score the third goal.
Lascelles, the club's captain, was forced at the break against Everton at St James's Park yesterday.
Everton were leading through Calvert-Lewin's 18th minute glancing header from Lucas Digne's cross when Pickford, failing to punch the ball clear, clattered into Rondon and gave away a penalty.
Referee Lee Mason, who endured a hard afternoon, awarded a penalty but no more, and the Magpies' frustration was compounded not only by Pickford's save from Ritchie, but by Richarlison's finish at the other end seconds later. However, their second-half showing could hardly have been more contrasting.
Newcastle players celebrate the winning goal
They had to wait until the 81st minute for the equaliser when Perez pounced as Pickford parried Miguel Almiron's long-range effort.
"This team is important", he said.
And the crowd went into raptures when Perez slotted home after coming back from an offside position to clinch an important three points.
The link between Ayoze Perez and Salomon Rondon, if Messi was doing that everybody would be talking about the goal.
With Pickford appearing to not play the ball, there could have been a debate over whether Lee Mason should have dismissed him, but he made the most of his fortune by keeping out Ritchie's penalty down the middle with his legs.