Robin van Persie, making his first appearance since Dec. 10 following a thigh injury, put the Red Devils in front after six minutes at Old Trafford before Ashley Young’s effort after the break wrapped up the victory.
The Premier League champions remain six points behind fourth-place Liverpool, and Moyes is keen for his squad to improve further in its bid to secure a Champions League berth.
"Don't think because I have won tonight I'm smiling and happy. With the vision I have got for the club we have got a long way to go," the Scot told reporters.
Juan Mata made his debut following his 45 million euro move from Chelsea, while Wayne Rooney returned from a groin problem to make an appearance off the bench against Cardiff.
But Moyes maintains the England striker and Van Persie will remain as a strike partnership in the wake of Mata’s arrival.
"I can play him [Mata] in any role. Left or right allows me to keep the Robin and Wayne partnership. I'm quite comfortable with that," said the United boss.