Carter barely looked troubled in reaching the mid-session interval 4-0 ahead. A 46 set the tone in the first frame and he returned to the table to finish the job.
A 58 break was enough to claim the second, with little in reply from Joyce, before Carter also took a slower third frame as Joyce struggled to find the form that saw him knock out world number one and defending champion Trump.
Carter made short work of the final frame before the break - although he missed the chance of a century, ending on 73 when the black jumped out of the bottom-left pocket.
When they returned, Joyce was handed a chance when Carter double-kissed a red and went to work, reaching 49 before playing safe. When Carter returned from an unexpected bathroom break, Joyce had a chance to seal the frame but missed a red when holding for the black and Carter came back to 57-39.
In the following safety battle, Joyce left a red on to the right middle for Carter, but he contrived to miss it and Joyce earned the frame with a lovely positional shot to the top cushion for the yellow before clearing to the pink.
And no sooner was he on the board than he won a second frame with a 76 break and the match was wide open again.
However Carter, beaten 9-6 in his only previous meeting with Joyce - in the first round of the 2010 UK - was quickly back into his stride in frame seven, moving 43-0 ahead before playing safe.
Joyce came back to 57-19 but went in-off seeking an unlikely route back - and Carter took the frame to lead 5-2.
He did not have to wait long for his chance to seal the win, knocking in a 73 break.
World number 50 Joyce, who had a brilliant amateur career - winning the European U19 Championship in 2001, the English Open in 2005 and the English Amateur Championship in 2006 - enjoyed his best run at a ranking event in reaching the last eight of the 2010 UK Championship. However he never looked likely to match that feat as he was outplayed for the most part by Carter.
Carter, twice a World Championship finalist - losing to Ronnie O'Sullivan in 2008 and 2012 - will face either Stuart Bingham or Stephen Maguire in the quarter-finals.
On table one, Stevens and Fu endured an attritional battle which contained some late brilliance from the Hong Kong cueman despite ultimately losing to the Welsh Dragon.
Fu looked out of sorts before the interval, overrunning his position on several occasions. World number 13 Stevens also started slowly, taking a scrappy opener then failing to capitalise on several chances as Fu somehow got over the line to level in the second frame.
After a messy start to the third frame, Stevens compiled a half-century with a disillusioned Fu opting not to play for snookers despite needing just three.
Fu - who is a prolific break-builder, having compiled 266 competitive century breaks during his career - was a shadow of the player who had compiled a tournament-leading 131 in beating Mark Allen.
With Stevens also struggling to take his chances, Fu was simply the least bad of the two in frame four, with barely any applause for the players as they left the arena with the score 3-1 to Stevens after a tedious 45 minutes of play.
When they returned, 2008 UK runner-up Fu missed an early chance to no one's surprise but Stevens was unfortunate to see a red drop after opening the pack up while potting the black. Fu was not on a red however and Stevens went on to move 4-1 ahead.
Fu came to the table in frame six a man transformed. He potted a difficult red after breaking apart the pack and seemed destined for his 27th century of the season, only to miss on 97.
An incredibly low-scoring frame followed, with Fu eventually 44-28 ahead with Stevens needing a snooker on the pink. Neither player wanted to take an unnecessary risk in so crucial a frame, and it contained some excellent safety play. Fu was finally able to pot the pink.
He then found himself 52 points down in the following frame, but a nerveless 60 clearance levelled the match at 4-4. With the momentum seemingly in his favour, he made a half-century in frame nine but missed a red on 52; Stevens worked his way around the table carefully, an impressive clearance ending a losing run of three frames.
Another half-century put him in the driving seat as he looked to clinch the match, and when Fu was unlucky to miss a long red down the right rail, he earned his quarter-final spot against either John Higgins or Mark Davis.
Stevens has won all five meetings with Fu, with three of those matches in the Championship League or PTC and the other finishing 10-3 in the first round of the World Championship earlier this year.