The England international started his fourth game in a row against the Toffees on Tuesday night but failed to find the net as the Red Devils slumbered to a 1-1 draw.
He told reporters: "Marcus Rashford hasn't scored a goal since September.
"The only thing he deserves is support. Nothing else. Support. No criticism. Support.
"He works, he works, he works. He tries, he tries. He plays through the middle, he goes to the left, he goes to the right.
"He tries, he tries, he tries. It's easy to see, the kid is desperate. But the kid is phenomenal - amazing professional, amazing professional.
"So no problem. I told him just now: 'No problem, keep going.'"
Mourinho has hit the headlines this week for his criticism of his own players, with Luke Shaw having been singled out after the Portuguese suggested the 21-year-old needs to change his football brain.