Defeat to Manchester United on Saturday left West Brom 17th in the table and only out of the bottom three by virtue of their superior goal difference over Cardiff City.
West Brom have only won once in their last 18 league matches and have yet to win under head coach Pepe Mel since he replaced Steve Clarke in January.
Anichebe, who arrived at The Hawthorns at the start of the season, remains confident they have the quality to stay in the division but issued a rallying call for everyone to pull together.
"The next couple of games against teams around us are the games that are important for us," he told the club's official website. "They're the games we need to start picking up points from.
"It's going to be important between now and the end of the season to win our individual battles with our opponents.
"We're in a serious position and we need to pull our socks up and get some points.
"It's down to us as players to go out there and play as well as we can.
"As long as we stick together we can get out of this."
Anichebe is the second-highest Premier League goalscorer at West Brom, with three goals to his name - bettered only by Saido Berahino's tally of four.
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