Despite dominating for long periods during the game, the Reds' breakthrough didn't arrive until just after the hour mark, when Robin van Persie fired home from the penalty spot following a foul on Patrice Evra.
Moyes says he expected a tough test against a determined Palace defence, but believes his side's performance was deserving of maximum points in south London.
"Crystal Palace, since Tony Pulis took over, has been a difficult place for anyone to come," Moyes told Sky Sports. "They haven't conceded many goals and they made themselves hard to beat again. Thankfully, we got two goals in the second half.
"In the Premier League, getting the first goal is always crucial. The way we played, we thoroughly deserved it. We had made some opportunities, and it was certainly deserved."
Wayne Rooney's unstoppable half-volley settled the contest with 20 minutes remaining, and moved him to within 40 goals of Sir Bobby Charlton's record as the club's all-time top scorer.
The United boss was full of praise for Rooney after the game, and he believes the striker can go on to make history at Old Trafford, after signing a contract extension earlier this week.
"The finish for the second goal was exquisite. Patrice [Evra] did really well to get to the byline and it was a great finish.
"He [Rooney] is a really important player. If you think about the amount of clubs who wanted to sign him, he would have been too difficult to replace.