The seamer, who has been left out of Australia's squad for this summer's eagerly-anticipated Ashes series, took five for 42 - to go with his five for 71 in the first innings - as Kent were all out for 135.
That left Northamptonshire a target of 104 at Canterbury for their third win of the season - a total they easily chased down to go clear at the top of Division Two.
It was also a day to remember for Copeland's compatriot Michael Klinger, whose unbeaten 103 propelled Gloucestershire to a nine-wicket thrashing of Leicestershire at Grace Road.
Set 188 for the first win of the campaign, Klinger and Chris Dent put on 129 for the opening wicket before the latter was trapped in front for 71 by Jigar Naik.
But Gloucestershire captain Klinger ensured there would be no collapse and hit the winning runs with his 16th four that also brought up his hundred.
Simon Kerrigan and Glen Chapple inspired Lancashire to an improbable 14-run victory over Glamorgan in a nail-biting final session on day three of their Division Two clash.
Slow left-armer Kerrigan took five for 32 while evergreen seamer Chapple chipped in with four wickets as the Welsh side - chasing 154 - collapsed to 139 all out after some lower order resistance from Lancashire had earlier seen them dismissed for 272.
Simon Katich top-scored with 65, while Kyle Hogg's 47 looks a precious innings now. Glamorgan seemed on course to canter to victory on 94 for two - largely thanks to opener William Bragg's 61 - but Kerrigan and Chapple had other ideas.