The 20-year-old, who has scored twice in the League Cup but has yet to open his account in the league, was fouled in the area by Brighton and Hove Albion defender Bruno in the final minute on Sunday, giving skipper Wayne Rooney the chance to slot home a penalty to tie the score at 1-1.
The young forward also had a chance to win the game in the dying seconds but failed to convert.
"I am getting a lot of chances in games. I want to take myself to the next level now and I need to start putting those away," he told Everton's website (www.evertonfc.com).
"I was playing back out wide. I had a couple of chances and was a bit frustrated not to score..."
Despite spending some 140 million pounds ($185.8 million) in the last transfer window, Everton have endured a woeful start to the campaign and sit 16th with eight points from eight games.
With Arsenal set to visit on Sunday, Everton will need a big performance if they are to start climbing the table and Calvert-Lewin said there were signs in the Brighton match that they were starting to show the necessary fight.
"We showed more character than we have in recent weeks to dig in late in the game and we will take that into the next one," he said. "We are creating chances and it is just a matter of putting them away now."
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Peter Rutherford)