The 28-year-old, whose England career has been affected by Chelsea's Ashley Cole being first choice in front of him for so long, opened the scoring with a fierce drive before adding his second from the penalty spot after Claudio Jacob fouled Victor Anichebe in the 2-1 win over West Brom.
Substitute Shane Long pulled one back midway through the second half after a mistake by John Heitinga but Everton had enough to see out victory and move to within a point of fourth-placed Tottenham.
"I think Bainesy has been playing really well and over the last couple of games I've been thinking he needs another challenge," said Moyes.
"He has to rise to it a little bit more and I think he has done that. He has picked himself up again.
"He has a lot of things going for him and he is turning out to be as good a left-back as there is.
"He keeps getting better when teams try to stop him, he tries to work on how he can alter doing things and he is a really good player.
"He is great for a manager because he wants to keep improving and we are very fortunate to have him here.
"He is carrying his ankle (injury) a bit but he misses no days of training and he gets on with it so he is a great player to have but he is also a great player to work with.
"He is a really good footballer, intelligent - apart from his style of hairdo I think he is all right."
There had been disappointment earlier in the day when the £8.6million deal for FC Twente midfielder Leroy Fer collapsed as the Dutch side refused to accept restructured payments after the player's medical highlighted an issue.
With one day remaining in the transfer window Moyes does not think he will have time to find a replacement at this stage.
"He is not our player but it is right we look after the boy and it didn't work for the reasons it didn't work," he said of the failed Fer deal.
"The boy is a very good player and we wish him well but on this occasion we just couldn't quite conclude the deal.
"There are lots of things connected to signing a player: it could be a contract, it could be the agent, it could be the clubs and medicals as well.
"On this occasion some of them didn't come together.
"It is very late now so I don't know if we will be doing anything."
West Brom boss Steve Clarke was left to bemoan his side's poor opening 45 minutes which contributed to extending their run to one point from a possible 15.
"If you go 2-0 down in the Premier League it is difficult to get a result," he said.
"We had a go in the second half but the key moment was the penalty on half-time - if we'd come in 1-0 down we would have had more chance of getting something from the game.
"I don't think we were great tonight [Wednesday] but I don't think Everton were particularly great either, it was a nothing game.
"We were quite comfortable until we gave away a bad goal but even after that I thought we were in a decent place and the penalty on half-time made it much more difficult."
Clarke has an injury concern over midfielder George Thorne, who was carried off with a knee injury in the first half and will be assessed on Thursday to discover the extent of the damage.
"Fingers crossed, but it doesn't look good," Clarke added.
Press Association Sport understands Everton will now focus their attentions on a deadline-day move for Sevilla striker Alvaro Negredo.
The 27-year-old, who has been linked with Tottenham this month, is a more advanced player than Fer so would not be a direct replacement for the Dutchman.
Reports have placed a potential fee at £13million, but with Sevilla experiencing financial difficulties Everton could pull off a cut-price deal for the Spain international.