The 24-year-old was almost the hero against Newcastle after replacing the injured Croatia international, scoring one goal and seeing another not given after the ball appeared to cross the line from a header.
His 88th-minute strike looked to have clinched the Toffees' third win in four Barclays Premier League matches only for Demba Ba's second of the night to secure a 2-2 draw for the Magpies.
Anichebe may find chances limited by the form of Jelavic, who injured a knee clattering into a post, and the summer arrivals of Steven Naismith and Mirallas, who has already impressed, but he remains upbeat.
"I am always happy just to be on the pitch as I've had some injuries in my time," he said.
"It is nice for me to learn off Niki as he plays as that lone striker and he's a great player.
"He plays between the centre-halves and that's what I'm trying to do more instead of dropping off.
"Kevin had some good cameo runs and he is one of those players who always wants to do something on the ball.
"He is a really good player and we now have great options.
"It is always nice to play and watch these kind of players as I am still learning.
"I've been working on that kind of turn and shot (for his goal) so I was happy with that but I'd have been happier with a win."
He was philosophical about not being awarded a goal earlier when the ball bounced down off the underside of the crossbar and went over the line before Mike Williamson cleared.
"Sometimes these things go against you but hopefully as the season goes on we'll get some luck," Anichebe added.
Mirallas had an all-action opening 45 minutes on his first league start and his performance bodes well for the future.
"He had three or four chances which he made himself," said manager David Moyes. "He will give us another attacking player who looks like he may have a goal in him.
"He scored a lot of goals in Greece and I think you could see that against Newcastle. Hopefully a few will go in for us."
Ba provided the perfect riposte to boss Alan Pardew having been left out in favour of Senegal team-mate Papiss Cisse.
The Magpies manager was pleased with the response and hoped other players in the same situation would do likewise.
"The fact Cisse got the goal for Senegal in the week gave him the nod because I was only going to go for one striker," he said.
"He (Ba) was aggrieved by that but he channelled it in the right way
"I have one or two others who are upset as well and I hope they channel it as well as they are going to get a game on Thursday (against Maritimo in the Europa League).
"Gabriel Obertan has been absolutely brilliant in training but he didn't yet a start and he was angry.
"I hope he comes on and gets two goals as well.
"It is hard having to leave good players on the bench but it gives everyone an opportunity to play in the first team."