Presented with a chance to open the scoring for United, the midfielder was guilty of missing an absolute sitter but made up for it in the second half when he found the bottom corner from the edge of the box.
Having seen an earlier chance pushed over the bar by Rob Elliot, the lively Anderson gave United the lead just before half time when his cooly struck effort went in off the post.
After the break United doubled their lead when Danny Welbeck teed up Cleverley and the England international found the bottom corner with his first United goal.
Alan Pardew responded to United’s second by bringing on Papiss Cisse and the Senegalese striker proceeded to cause trouble for the hosts immediately. He ghosted into the United box to head Newcastle back into the game in the 68th minute and generally made a nuisance of himself as Newcastle searched for a second.
He almost grabbed a spectacular equaliser when he chested down and a cross and overhead kicked an effort onto David De Gea’s crossbar, but ultimately it would be United that would progress to the next round, where a trip to Stamford Bridge awaits Sir Alex’s men.