Striker Dominic Calvert-Lewin was sent off for the first time in his career but Everton's 10 men held on to earn an FA Cup third-round replay with Crystal Palace following a 0-0 draw at Selhurst Park.
It proved a costly night for the Toffees, who are set to be without their forward for three matches, and had Dwight McNeil carried off late on.
Speaking after the match Everton manager Sean Dyche said: "It's early but Dwight's in the dressing-room and calm, which is a good sign. Hopefully it's a swollen ankle, but it's too early to tell."
How the match unfolded
Palace handed a first senior start to 19-year-old Brazilian forward Matheus Franca in the absence of the injured Michael Olise, with Jeffrey Schlupp also coming in to their starting XI. Joao Virginia replaced Jordan Pickford in goal, one of three changes made by Everton.
Both teams were frustrated in a first half that produced plenty of effort, but not much finesse in the final third.
Arnaut Danjuma's 18th-minute shot deflected off Chris Richards and flew into the side-netting and Calvert-Lewin headed wide under pressure from Marc Guehi with Everton's best opportunities.
At the other end, Eberechi Eze teed up Jefferson Lerma, who fired narrowly over, while a superb sliding tackle by Seamus Coleman on Jean-Phillipe Mateta just diverted the ball through to goalkeeper Virginia to avert the danger moments before the break.
Everton upped the tempo in the second half and looked the more likely to score. Calvert-Lewin, without a goal in 12 matches, fired a tame shot straight at Dean Henderson, Amadou Onana headed over, and Danjuma's effort was comfortably held by the goalkeeper.
However, Everton were forced to play the final 11 minutes with 10 men as referee Chris Kavanagh deemed after a that Calvert-Lewin's foul on Nathaniel Clyne was worthy of a red.
Dyche responded by bringing on Beto, and the substitute nearly produced the opening goal six minutes from time. Beto found James Garner in the box, but the Toffees midfielder was leaning back and fired over from close range.
Everton were forced into another change with two minutes of normal time remaining when McNeil appeared to collide with team-mate Jarrad Branthwaite when defending a Palace corner.
There was still time for Lerma to lash an effort narrowly over and Virginia to push away an Eze effort at full stretch as Palace tried without success to make their numerical advantage count.
They will face each other once more in the third-round replay on Wednesday 17 January. Everton are next in Premier League action on Sunday 14 January at home to second-place Aston Villa, with Palace visiting Arsenal the following Saturday.
Both Palace and Everton will be in the FA Cup fourth-round draw which will be made prior to Manchester United's tie at Wigan Athletic on Monday.
See: Crystal Palace report and line-up
See: Everton report and line-up
FA Cup action continues
The action continues on Friday night with three matches, including two all-Premier League ties, starting with Wolverhampton Wanderers' visit to Brentford less than two weeks after beating the Bees 4-1 in west London.
Tottenham Hotspur host Burnley, having already beaten the Clarets 5-2 at Turf Moor in the league this season, while Fulham entertain a Rotherham United side bottom of the Championship.
Friday's FA Cup third-round matches
19:15 Brentford v Wolves
19:30 Fulham v Rotherham
20:00 Spurs v Burnley
