A Chris Cohen free-kick had given the Championship side a first-half lead before the £2.65million capture from Derby struck twice after the break - either side of a well-taken goal from young striker Adam Newbold - to put Forest through.
Morecambe had knocked out both Preston and Wolves in last season's competition, but a giant killing was rarely on the cards at the City Ground.
In the first five minutes an arching 40-yard pass from Ian Breckin sent full-back Luke Chambers racing into space down and he delivered an equally accurate right-wing cross to the near post where Arron Davies flicked a header onto the post.
A controversial goal gave Forest the lead after Lee Martin - the lively new loan signing from Manchester United - had been booked for taking a free-kick too quickly on the edge of the box.
Referee Carl Boyeson ordered it to be taken again and Cohen duly bent the ball into the bottom right corner of the net in the 14th minute.
Martin was involved again as he slid Earnshaw through on goal, only for former Forest keeper Barry Roche to deny him with a smart stop.
Fraser McLachlan rifled over as Morecambe enjoyed a rare opportunity, while Earnshaw and Garath McCleary were both off-target for Forest following two flowing attacks.
And Roche ensured Morecambe went into the break only a goal behind as he smothered another effort from Earnshaw when the striker seemed certain to score.
Earnshaw made no mistake in the 61st minute though when, having been sent racing clear by defender Joe Heath, he rolled the ball calmly inside the far post.
And after long spells of pressure, Newbold netted the third in the 81st minute by rifling high into the back of the net from 12 yards.
Earnshaw then doubled his tally eight minutes later by rolling into the same corner of the net after charging clear again.