Black Cats boss Moyes has been under pressure following their poor start to the season, but Anichebe inspired them to a first victory of the campaign at Dean Court.
Moyes handed the striker, who he worked with at Everton, his first Premier League start for the club and Anichebe repaid him with the equaliser to cancel out Dan Gosling's opener.
Sunderland were reduced to 10 men when Steven Pienaar was sent off for a poor challenge, but Anichebe won a late penalty which Jermain Defoe dispatched to secure a long overdue three points for the bottom side.
"It was a g reat performance by the team, and it was really important to get the three points," Anichebe told the club website
"When we went down to 10 men the fans were like the 11th man for us, they really pulled us through so credit to them, and credit to the manager.
"The manager has a lot of pressure on him even though it's not him going out to play
He's a really good manager but he just needs time to put down what he wants us to do.
"There were signs of what we used to do back at Everton, we pressed, showed resilience and dug deep and that's the most important thing
That's the foundation, and that's what we did
It's been coming.
"So credit to the manager
We really pulled out a result for him and long may it continue.
"Where we are at the moment is not where Sunderland should be, it's criminal really, but hopefully we can get some players back and continue to do well."
Bournemouth have now lost two on the trot and will be desperate to bounce back when they head to Stoke after the international break.
"It's going to be a long two weeks now," Gosling told the club website
"The last two games have been disappointing.
"We were in the top 10 and in a great position to kick on from there and get ourselves higher up the league.
"But it wasn't to be and now we need two weeks of hard work to get back to winning ways in the next game."
Source : PA
Source: PA