Anichebe revealed the Hawthorns academy graduate idolises Tottenham and England forward Defoe – and has now been nicknamed 'Defoe' by the former Everton man.
Berahino, who this week signed a new three-and-a-half-year Albion contract, has burst onto the first-team scene this season with a total of 12 goals for the club and England Under-21s.
Anichebe said: "He's a really good player and a great finisher. He's a young player with a lot of confidence and he's come into a league where nobody really knows him.
"And he just enjoys it. It's nice to see someone that just comes in every day and enjoys it. Sometimes with older players it's more of a job, but he has come from nothing and he's come in and taken his chance.
"He's an unbelievable finisher. I call him 'Defoe' because that's how he is.
"He loves it. He calls himself 'Defoe'. He follows him on Instagram and he thinks he's Defoe!
"He's a confident boy but I'm delighted for him that he's got a new deal."
Anichebe, who opened his own account for Albion against Manchester City on Wednesday, has experience of emerging from the youth ranks, having broken through at Everton as a teenager.
And the Nigeria international believes Berahino will need to be patient as he awaits regular starts.
"It's difficult for him," said Anichebe. "I speak to him sometimes because there are a few games when he's come on and scored and they he's got frustrated because he's not started the next game.
"But every young player goes through that. When I was at Everton I'd come on and score and then not start next game, but every young players goes through that.
"You just play your part in the team when there are senior players in front of you and you just wait your turn.
"He has waited his turn and he's come in and done well."