Calvert-Lewin headed in from an Ademola Lookman delivery in the 87th minute, five minutes after both players had come off the bench.
Tosun, who had come on midway through the second half, then added an 89th-minute strike.
Toffees and England goalkeeper Jordan Pickford had earlier kept the score at 0-0 by saving a Luka Milivojevic penalty on the hour mark.
Palace, who have scored only five goals in nine league matches this term, went close in the first half when Cheikhou Kouyate's header hit the bar.
Everton, having built on wins over Fulham and Leicester, are now up to eighth in the table, while Palace have dropped a place to 15th after their third consecutive defeat.
After a slow start Palace started to show more threat, and after James Tomkins headed a Townsend cross wide and Pickford turned behind Milivojevic's free-kick, Kouyate saw his headed effort come back off the bar.
Second half penalty saved
Palace were handed a golden opportunity to break the deadlock shortly after the break when Coleman brought down Wilfried Zaha in the area and referee Anthony Taylor pointed to the spot.
However, the visitors were unable to do so, with Milivojevic's strike from 12 yards - sent down the middle - being kept out by the diving Pickford's right foot.
Moments later, Everton failed to take a great chance themselves as Theo Walcott, played in one-on-one, saw his shot kept out by Hennessey's feet.
Toffees boss Marco Silva brought on Tosun for Bernard, and after debutant Andre Gomes then curled an attempt wide, Calvert-Lewin and Lookman were also introduced.
All three substitutes then made their mark in a dramatic finale as Calvert-Lewin converted from Lookman's cross, before Tosun then latched on to a long ball forward from Keane and drilled a low shot in.
Everton v Crystal Palace Opta stats
- Everton have won three straight Premier League matches for the first time since January 2017.
- Crystal Palace are now winless in four Premier League matches (D1 L3), their longest such sequence since March (run of seven games).
- Crystal Palace manager Roy Hodgson is now winless in all eight of his Premier League visits to Everton (D1 L7), against no other side has he managed more away games in the competition without winning at least once (also eight v Tottenham).
- For the first time since August 2017 (Man Utd v Leicester – Rashford, Lingard and Fellaini), three substitutes were directly involved in a goal in the same Premier League game (Calvert-Lewin, Lookman and Tosun for Everton today).
- Everton have scored more headed goals in the Premier League this season than any other side (six).
- Dominic Calvert-Lewin and Cenk Tosun are the first two Everton substitutes to score in the same Premier League game since March 2014, when Kevin Mirallas and Steven Naismith netted against Fulham.
- Luka Milivojevic’s penalty miss was his first in the Premier League for Crystal Palace since December 2017 v Manchester City, and just the second penalty he has failed to convert in 11 attempts in the competition.
- Since the beginning of last season, only Manchester City’s Raheem Sterling (six), has won more Premier League penalities than Crystal Palace’s Wifried Zaha (five).
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