Gianluca Scamacca opened the scoring on his return to the starting line-up, with Jarrod Bowen adding a quickfire brace early in the second half, after the Cypriot side were reduced to 10 men just before the break.
Youngster Divin Mubama set the seal on a 4-0 victory at London Stadium, after Michail Antonio’s brace had earned the Hammers a 2-0 win in the first leg.
And Moyes said: “Really pleasing. I don’t think it’s something that happens all the time.
“We’ve had two quarter-finals in the last two years in Europe, we’ve had some great trips with the supporters, played in some brilliant stadiums, and it’s been great for the club, it really has been.
“It’s been great for my staff and the players, we’ve really enjoyed it.
“I thought we started the game really well and probably should’ve scored before we did do.
“But when we did get the goal it was good and we saw the game off and the sending-off changed the game as well.”
Italian striker Scamacca had not started a game since the January 4 game at Leeds, but took a pass from Manuel Lanzini to stab it underneath the goalkeeper midway through the first half.
And when his low drive was saved after the restart, Bowen pounced to double the lead, before looping a header inside the far post moments later.
Moyes added: “He [Scamacca] did ok, got a goal, a wee bit fortunate the goal, but strikers need goals to gain confidence and I’m pleased for him.
“He’s a really good lad and done a lot of good things tonight.
“Jarrod’s a huge player for us, creating chances, creating goals and scoring goals.
“I thought his first goal was a great finish and the second one was a deft header so really good for Jarrod. I thought he played really well on Saturday and to back that up with two goals.”
Mubama had been denied a first senior goal during the group phase, but showed a predatory instinct to flick Tomas Soucek’s header from an Aaron Cresswell corner into the net from close range.
And Moyes was particularly pleased to see the 18-year-old open his account, adding: “There’s nothing gives me more pleasure than giving young players an opportunity.
“It’s a really difficult road to become a professional footballer, especially in the Premier League. All of these boys are starting at nine, 10 years old, trying to get through.
“To be able to give them a chance, tonight the result allowed us that opportunity as the game went on.
“Every time Divin has come on, his runs, his work, he doesn’t think twice about his effort because of that we’ve grown to him all the time. He’s done really well.”
West Ham are joined in the last eight by KAA Gent, Lech Poznan, Fiorentina, Basel, AK Alkmaar, Nice and RSC Anderlecht and Moyes wants the journey to continue.
“We’ve had some incredible nights. This time last year [in the Europa League] it was Seville, then Lyon and Frankfurt. I hope we can keep the journey going and go a step or two further,” he said.
“This competition now, we’ve given ourselves a real chance and we’ll all be excited for the draw and it’s great to be in another quarter-final.
“It’s something that is very hard to do and to do it two years in a row is very good.”
The Hammers now go into the international break, with their scheduled Premier League fixture against Manchester City postponed due to the FA Cup quarter-finals.
They do not return to action until Southampton visit on April 2, but Moyes knows he has little time to rest, with his side sitting just outside the relegation zone on goal difference at the moment.
“I’ve got a few teams to watch to get myself ready for what we all know is going to be a colossal couple of months,” he added.
“We’ve got a huge amount of games and also the level of importance.”
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