Liverpool keeper Loris Karius also had to be at his best to deal with a curling shot from the otherwise ineffectual Yannick Bolasie although Everton seldom showed the forward momentum to threaten his goal.
Wayne Rooney was clearly unhappy to be substituted for the fit-again Idrissa Gueye early in the second half but the impetus remained with Liverpool, who brought on Alex Oxlade-Chamberlain and Firmino, until a frantic final 10 minutes in which Seamus Coleman and Dominic Calvert-Lewin went close to grabbing an unlikely winner for the home side.
Both keepers were predictably happy with not conceding in a game in which both delivered assured performances.
Karius, who faces constant speculation that Liverpool will replace him in the summer, said he did not feel under pressure to keep his place.
"I don't feel on trial. If I believe everything I read we will have 10 goalkeepers at the club next season," he said.
"I was just happy to keep a clean sheet and get my fingertips to the shot (from Bolasie)."
On a slippery pitch, Pickford also impressed in front of watching England manager Gareth Southgate. "It was tough conditions for us keepers. If I keep a clean sheet I am happy."
Everton manager Sam Allardyce defended his decision to take off Rooney, which he said was down to the need for "fresh legs" in midfield.
"I make the decisions and I thought we did the right thing to make those changes to give us a better chance," he said.
The result means that both league games between the old rivals have been drawn this season following the 1-1 at Anfield.
Tottenham Hotspur can displace Liverpool in third place if they beat Stoke City by five goals or more later on Saturday.
(Reporting by Neil Robinson; Editing by Clare Fallon)
