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.
Everton missed two great chances to end a 16-match winless run against Liverpool as they drew 0-0 in the Merseyside derby at Goodison Park.
With a top four place nearly wrapped up for Liverpool and with a crucial Champions League second leg tie against Manchester City coming up mid-week, Klopp had a choice about what game he wanted to prioritise.
Klopp understandably had one eye on Tuesday in his selection, and Liverpool's attacking threat was considerably blunted without Mohamed Salah and Roberto Firmino in the starting line-up.
Pickford, who was watched by England manager Gareth Southgate, said: "The first feeling is we got the clean sheet which we've been drilling for a while". It was like a different Everton team for the last 15 minutes.
The home side might even have won the match at the end, as January signing Cenk Tosun nodded Theo Walcott's cross towards the far post, where Coleman just failed to turn the ball in.
Calvert-Lewin blasted off target one minute later as the spoils were shared, doing little for either side - although Liverpool will be delighted to avoid any injuries ahead of a massive game against City on Tuesday. It was only during the last 10 minutes when Everton had a late surge, producing a number of opportunities which they failed to convert.
Liverpool were prepared to show aggression, with Ings involved in a confrontation with Coleman that needed the intervention of referee Michael Oliver and both captains to sort out.
For Liverpool James Milner was peppering the goal area with crosses and shots - with Klavan behind him he had plenty of freedom to get himself forward.