After mopping up Kent's first innings for 236 in the morning session, despite Sam Northeast racking up a century, Durham had a 212-run lead.
Graham Clark's career-best 83, helped by skipper Paul Collingwood's 51, saw Durham declare with a lead of 483.
Kent survived the final 10 overs of the day to end of 35 without loss.
Northeast had been the Kent hero on day two, with his unbeaten 109 aided by Yasir Shah's hard-earned 48 in a 113-run eighth-wicket partnership, before Durham hit back to reduce Kent from 236-8 to 236 all out.
South African opener Stephen Cook (44) continued his resurgence in Durham colours once he took to the crease, although partner Keaton Jennings was not so lucky as he departed for a first-ball duck.
Cameron Steel, who made 43, along with Clark and Collingwood put Jon Lewis' side in a decent position at the declaration, in spite of early exits for Ryan Pringle and Stuart Poynter.
Daniel Bell-Drummond and Sean Dickson nursed Kent to stumps without loss, but still trailing by 449 runs.
Durham put themselves in strong position to earn a first Championship win of the season after declaring with a big lead on the third day at Kent.