Calvert-Lewin saw red after a VAR review of the England striker’s studs-up foul on Nathaniel Clyne late in the second half at Selhurst Park.
Calvert-Lewin made only slight contact with Clyne’s leg while winning the ball and Everton boss Sean Dyche was furious with the decision by referee Chris Kavanagh after a long VAR check.
The first red card of Calvert-Lewin’s career could have been fatal to Everton’s hopes of progressing in the competition.
They also lost winger Dwight McNeil to a serious-looking injury in the closing stages, but Dyche’s team held on through nine minutes of stoppage-time to force a replay at Goodison Park later in January.
Five-time FA Cup winners Everton last lifted the trophy in 1995 and have been searching in vain for another piece of major silverware ever since.
Hit by a 10-point deduction for financial breaches earlier this season, fourth bottom Everton are mired in a Premier League relegation battle.
An FA Cup run would be a welcome distraction from their league travails and they can still dream of the road to Wembley thanks to this gritty display.
Everton won 3-2 at Selhurst Park in the Premier League in November, but after losing their previous three games they happily settled for avoiding defeat this time.
Deputising for the rested Jordan Pickford, Everton keeper Joao Virginia had to make an early save to repel Jeffrey Schlupp’s half-volley.
Everton responded with an incisive raid as Calvert-Lewin teed up Arnaut Danjuma for a shot that deflected off Chris Richards into the side-netting.
They went close again when Jack Harrison took Danjuma’s flick and broke clear for a shot that forced a fine save from Dean Henderson.
Henderson had to be alert moments later, saving with his foot after James Garner’s corner nearly sneaked in at the near post.
Schlupp’s cross was cleared off the line by Vitaliy Mykolenko before Palace’s Jefferson Lerma smashed narrowly over from the edge of the area.
Eberechi Eze threatened to break the deadlock early in the second half, but the Palace winger’s effort was turned behind by Virginia, who clutched Lerma’s header from the resulting corner.
Calvert-Lewin is without a goal since October and the Everton striker’s lack of confidence was clear when he shot tamely at Henderson from a dangerous position.
Presented with a good chance of his own, Danjuma fared no better than his team-mate as his strike failed to extend Henderson.
Calvert-Lewin got his marching orders in the 79th minute following his debatable foul on Clyne, but Everton stood firm to ensure no more damage was done.
AFP
