With 10 fixtures left to go this season, the Baggies are currently 17th in the Barclays Premier League and separated from the relegation zone by goal difference alone.
They have not won any of their seven matches under Mel following his appointment in January and reports have suggested the Spaniard's job is under threat.
Anichebe is in no doubt that things need to change - but on the pitch, in terms of the players' performance.
"We're in a serious position and we need to pull our socks up and get some points," Anichebe, quoted by West Brom's official website, said.
"The next game (Saturday's trip to Swansea) we need to improve and all of us know that.
"It's down to us as players to go out there and play as well as we can.
"It's not all down to the head coach. We, as players, need to go out and do well for the team and the supporters.
"Whatever team the head coach puts out, we're the ones who step over the line and we need to work hard and fight."
West Brom's last outing saw them beaten 3-0 at The Hawthorns by Manchester United on Saturday. After Swansea, they face Hull and Norwich away, either side of hosting Cardiff.
Anichebe said: "We can't be too down about last Saturday because we're not going to play Manchester United every week.
"We've had some good results against top teams, but the next couple of games against teams around us are the games that are important for us.
"They're the games we need to start picking up points from."
Albion have not kept a clean sheet since January 1, and the 25-year-old Nigerian added: "The most important thing is we need to be really hard to beat.
"It doesn't matter about scoring one, two or three goals.
"The most important thing is keeping a clean sheet and being hard to beat then try to nick a goal or give our players the chance to do something special."