The Hammers are currently just a point off the top six following a 2-0 win at home to Manchester United last weekend.
And the Scot is confident in West Ham's ability to get better throughout the season.
"We're really pleased with the progress we've made," Moyes told a pre-match press conference on Wednesday.
"We've got 30 points in the league, which is a really good total for us at this time of year. We want to keep it going and we think we can get better and do even more."
Moyes added striker Michail Antonio will again be absent as he continues his recovery from a knee injury, with centre-back Nayef Aguerd to undergo assessment after missing the United victory because of illness.
West Ham defeated Arsenal 3-1 in the English League Cup last month but Moyes is prepared for a far tougher test on Thursday.
"(I'm expecting) huge challenges. Arsenal have always been a really hard team to beat," he said. "We had a good result against them in the cup but this is a different competition.
"The league throws up different players and Arsenal changed their team for the game in the League Cup so they will be back to their full-strength team, I've no doubt, and going to the Emirates Stadium has never been an easy game."
Arsenal will go top of the table if they beat the Hammers and Moyes praised the "brilliant job" done by Gunners counterpart Mikel Arteta.
"I've been impressed with Mikel since he first went in," Moyes said.
"He won the FA Cup and the Community Shield, so a couple of wins right at the start.
"Then obviously it went a bit difficult and there was a lot of talk about him, but he hasn't half proved all those doubters wrong because he's done a brilliant job with Arsenal. He's got them incredibly competitive, back around the top of the league and competing again.
"I think most Arsenal supporters will be happy with Mikel."
