De Gea was heavily criticised when he moved to United three years ago, but the Spaniard is now regarded as one of the top goalkeepers in the Barclays Premier League.
De Gea underlined his importance to the club when he pulled off a magnificent double-save to keep the score at 1-0 during Wednesday's 3-0 win over Olympiacos.
Moyes was delighted with the 23-year-old's performance in the crucial win and he thinks the goalkeeper will get even better.
"David de Gea has been terrific," Moyes said.
"He has been great. He came in as a good young goalkeeper and he is turning into a great goalkeeper with a lot more to come."
There have been suggestions that Barcelona will launch a bid to sign De Gea this summer.
The stopper has two years left on his current deal and Moyes gave a positive response when asked if he thought the club would be extending that contract.
"The club are always going to keep their best players," the United manager said.
"The club will always look to keep the players who they feel they need to."
De Gea will no doubt be called into regular action in the next round when United face holders and tournament favourites Bayern Munich.
The Spaniard admits the draw could have been easier for the Red Devils.
"You've got to be honest and say we haven't been too fortunate in the draw," De Gea told United Uncovered.
"I suppose everyone would agree that they probably are the strongest outfit left in the tournament."
Bayern had little trouble defeating Arsenal in the previous round and Pep Guardiola's men are currently unbeaten in the Bundesliga.
But, after being part of the team that came back from a two-goal deficit against Olympiacos, De Gea thinks an upset could be possible.
"You're going to get a tough team at this stage, but the main thing is that we go there with confidence now," De Gea said.
"We must fight really, really hard and really go for it. That's what we've got to do. One team's got to win."
Among Bayern's stellar squad are two players De Gea knows well - his international team-mates Javi Martinez and Thiago Alcantara.
"It will be great to see them both here and in Germany for the away leg," De Gea added.
"I get on really well with Thiago and Javi, we have got a fairly close relationship. It will be great to see them but let's hope, as great as they are, that they're not quite on top form when it comes to playing in those two games."