United went into Tuesday's tie against Shakhtar Donetsk needing only a draw to secure top spot in Group A, but went one better as Phil Jones struck in the 67th minute to earn a 1-0 win.
Moyes is adamant a club of United's size needs to be aiming to win every trophy available and was relieved to overcome Shakhtar following a poor first-half performance.
The Scot told ITV: "A club like Manchester United has to be trying to win the trophy, that's their job. You have to get better as the tournament goes on. It's a tough competition, a hard one, but we will take it one game at a time.
"As a club we could be considered one of the favourites, but as a team we have to improve if we are going to keep progressing. Hopefully we have given ourselves the best chance by winning the group.
"We wanted to top the group and we have. The win helps us, improves our form. We didn't play well for 30 minutes and they (Shakhtar) had two or three opportunities.
"In the first half they had the better chances. In the second half it was much better. A loss would have been disappointing. I was tough with them (United's players) at half-time but we needed to be. We need to win games."
Jones acknowledged United, who are currently ninth in the Premier League, were lacklustre in the first period, but noticed a big improvement after half-time and praised the impact of the fans at Old Trafford.
He added: "We needed to show a bit more desire [in the first half], a bit more belief in our performance and we did that eventually.
"In the first half we were shaky and tentative but in the second half we got to grips with things and created more chances.
"People have opinions [about the club's start to the season], but the fans were excellent tonight. They stuck by us and hopefully we can start a run and kick on from here."
