Tom Cleverley then missed a simple chance from 12 yards, sliding a side-foot effort just wide of the post.
United did take the lead just before the interval when Anderson held off two Newcastle challenges before rifling in off the post from 20 yards.
Cleverley made up for his first-half miss 13 minutes into the second period when he curled home from the edge of the area for his first goal in a United shirt.
The hosts almost got a third minutes later when Javier Hernandez broke clear of the Newcastle defence but the Mexican's powerful strike slammed against the face of the crossbar.
Newcastle pushed for an equaliser in the final 15 minutes and came close, with first Dan Gosling watching his curled effort sail just over before Cisse rattled the frame of the goal with an overhead kick.
The draw for the fourth round is due to be made shortly.