The left-hander registered breaks of 100 and 67 twice and while King scored well when presented with an opportunity, they were hard to come by as Trump advanced to a fourth-round clash with Joe Perry.
Stuart Bingham registered the highest break of the tournament so far en route to a 6-2 victory over David Gilbert.
The former world champion started the match with a 145 clearance as each of the first four frames of the match produced a century, Bingham making 102 and 108 in frames two and three with Gilbert's 100 sandwiched between them.
Bingham edged two tight frames to move within one of the final and while Gilbert produced another excellent run, his break of 91 taking frame seven, Bingham soon wrapped up victory.
Jack Lisowski beat Marco Fu 6-3, a tight eighth frame proving key. Fu had made a 119 break in the sixth and then added the seventh, but the chance to draw level went begging and Lisowski went on to win comfortably.
Neil Robertson edged past Graham Dott 6-5 to book his place in round four against fellow left-hander Mark Allen.
Dott was fastest out of the blocks and raced into a 2-0 lead courtesy of breaks of 66 and 58.
Robertson responded with a pair of 50 breaks himself to level the scores at 2-2 before the next four frames were shared.
A well-taken break 68 put Roberston ahead for the first time in the match and though Dott fought back to force a decider, it was the Australian who held his nerve to come out on top.
Next up is Northern Ireland's Allen, who won the first three and last three frames in a row to beat Hossein Vafaei 6-2 courtesy of three breaks in the 80s.
Two-time UK Championship winner Ding Junhui beat compatriot Xiao Guodong in an entertaining 6-4 victory that saw Xiao hit breaks of 126 and 100 to come from 3-0 down to level at 3-3 - but Ding held his nerve with back-to-back frames later clinching the match.
Afternoon session (1300 GMT)
- Marco Fu 3-6 Jack Lisowski
- Judd Trump 6-2 Mark King
- Stuart Bingham 6-2 David Gilbert
- Neil Robertson 6-5 Graeme Dott
Evening session (1900 GMT)
- Xiao Guodong 4-6 Ding Junhui
- Sunny Akani 6-2 Jak Jones
- Gary Wilson 4-6 Barry Hawkins
- Mark Allen 6-2 Hossein Vafaei
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