The 36-year-old free agent, who has previously spent eight-and-a-half years at Manchester United, was signed on a short-term basis earlier this week.
Moyes knows Evra well from their short time together at Old Trafford and insists that the Frenchman will provide more than just experience.
"We like his other qualities but we got him here to play. Hopefully we can get him ready as quick as possible," he told reporters at his pre-match press conference. "I know Pat well and I know what he is capable of. I know his leadership qualities and what he will bring to the dressing room.
"Also his level of professionalism is as good as anybody. He's seen and worked with the best so hopefully he can pass that on to the players. We needed him after Arthur's suspension.
"If Aaron Cresswell got injured we would be left short. We were so busy with other players we didn't do it in January. When the window closed and it was a bit calm we could get it done.
"There's a chance [he could play against Watford]. He's not too far away, he has been training. He played in November for Marseille so he's no different from someone who may have had a long 10-or-12-week injury. Sometimes getting back playing quickly is the best thing to do."
Evra was sacked by former club Marseille for kicking a supporter ahead of a Europa League tie with Vitoria Guimaraes.