The English 16-time world champion beat the Belgian 4-2 at Alexandra Palace.
Taylor will meet five-time world champion Van Barneveld for the fourth time at the tournament.
Dutch top seed Michael van Gerwen beat England's Darren Webster 4-1 while Scottish third seed Peter Wright beat England's Ian White by the same score.
Taylor battled back from behind in each of the opening three sets to lead 3-0, only to see 13th seed Huybrechts narrow the gap to 3-2.
But the fourth seed survived the spirited comeback, sealing victory at the second opportunity in the sixth set to tee up the meeting with long-time rival Van Barneveld on Friday.
Taylor has won two of their three previous encounters at the event, with Van Barneveld prevailing 7-6 in an epic 2007 final.
Fellow Dutchman Van Gerwen was pushed by Webster - who was the lowest-ranked player left in the competition - in the first two sets, then took the third without dropping a leg.
Webster, 46 from Norwich, arguably raised the biggest cheer of the night when he held off the Dutchman to win the fourth set.
Both players came close to nine-dart checkouts in the fifth set, with Van Gerwen punishing his opponent's inability to complete the feat by sealing victory with his next throw.
In Thursday's afternoon session, sixth seed James Wade twice fought back from two-set deficits to beat fellow Englishman Michael Smith 4-3.
Smith raced into a 2-0 advantage and then led 3-1 before Wade clinched victory by winning seven straight legs.
England's Dave Chisnall reached the quarter-finals for the first time as he held off Dutchman Jelle Klaasen to earn a 4-2 victory.
Northern Ireland's Daryl Gurney also progressed to his maiden last-eight appearance after edging past Welshman Mark Webster in a thrilling 4-3 win.
Daryl Gurney (NI) 4-3 Mark Webster (Wal)
Dave Chisnall (Eng) 4-2 Jelle Klaasen (Ned)
James Wade (Eng) 4-3 Michael Smith (Eng)
Peter Wright (Sco) 4-1 Ian White (Eng)
Phil Taylor (Eng) 4-2 Kim Huybrechts (Bel)
Michael van Gerwen (Ned) 4-1 Darren Webster (Eng)
Friday's afternoon session (13:00 GMT)
James Wade (6) v Peter Wright (3)
Gary Anderson (2) v Dave Chisnall (7)
Friday's evening session (19:00 GMT)
Daryl Gurney (24) v Michael Van Gerwen (1)
Raymond van Barneveld (12) v Phil Taylor (4)

The Gran Fondo was in the city for the first time, giving riders the option of taking on a 100 mile (160km) course or a shorter 37 mile (60km) route.
Plymouth based cyclist Jonathan Tiernan-Locke, who is serving a two-year doping ban, was among the riders.
The Sky Ride offered families the chance to cycle a 5 mile (8km) tour of the city.

Police were called to an address in Holland Street at about 16:15 on Wednesday where they found the body of James Chadwick, who was from the city.
He had a number of unexplained injuries.
Following the results of a post-mortem examination, his death is now being treated as murder.
Officers have been carrying out door-to-door inquiries and reviewing CCTV images from the area.
Det Ch Insp Iain Smith, of Police Scotland, said: "This investigation is at an early stage and it is important we establish what has happened which led to James Chadwick losing his life.
"Our inquiries so far have established that Mr Chadwick was last seen on Monday 31 August and we're appealing to anyone who has any knowledge of his movements since 31 August to contact police as a matter of urgency."
Phil Taylor set up a tantalising PDC World Darts Championship quarter-final against Raymond van Barneveld with a hard-fought win over Kim Huybrechts.
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