The 20-year-old had come in for some criticism following his £18m move from Atletico Madrid over the summer, but recent performances against Chelsea and Stoke have convinced Evra that the Spaniard has great potential.
"David is doing very well - his confidence is going up all the time," Evra told MUTV.
"You can see when he makes a great save, all the players are very happy for him. We are all behind him and support him.
"I am really confident in him. I am looking forward and he can become one of the best."
United continue their Champions League campaign on Tuesday when they take on Basel at Old Trafford.