The East London side made the semi-finals in 2022, where they were eliminated by eventual champions Eintracht Frankfurt (3-1), before they triumphed in the third-tier Europa Conference League after beating Fiorentina 2-1 last year.
"We've been on a really good run over the last few years, so we're not coming here without expectancy. We've done very well in Europe over the last three years," Moyes told a press conference on Wednesday ahead of the first leg.
"We're looking forward to the game. It'll be difficult, but there are two legs and we're hopeful.
"We have character, and it's probably a bit about my character as well.
"We'll need leadership, composure and tactical understanding in this tie, against a team who are unbeaten in 41, so we need to try and handle that."
Leverkusen have won all but five games across all competitions in 2023-24, having equalled a league record unbeaten run of 28 consecutive matches when they battled past Union Berlin 1-0 on Saturday.
"It could be the toughest challenge yet. They haven't lost for a long time, and we know how well they have played this season," said West Ham defender Kurt Zouma.
"But, we are still confident. We know the qualities we have and we showed that against Freiburg. It's two legs, and hopefully we can get a good result tomorrow."
West Ham beat Freiburg 5-1 on aggregate in the last 16 and Leverkusen advanced 5-4 past Azerbaijan's Qarabag after both sides topped their respective groups.
The visitors will face Leverkusen without forward Jarrod Bowen, midfielder Kalvin Phillips and goalkeeper Alphonse Areola.
"Jarrod isn't with us, he's twisted his knee and there's some sort of damage. We can't give any dates of when he might be back, unfortunately," Moyes said.
"Kalvin Phillips has picked up a problem and will be missing tomorrow. It looks like he's tweaked his hamstring a bit, but we don't think it's a long-term problem."
