Sussex, who began on 11-1, were in danger of following on after Wayne Parnell (3-48) and Mitch Claydon (4-87) struck to leave them 97-6 after lunch.
But Brown (90) and Archer (60 not out) frustrated Kent as the hosts made 291, 13 runs short of Kent's total.
In reply Kent lost Daniel Bell-Drummond (35) and Joe Denly to Ajmal Shahzad (2-15) and reached stumps on 82-2.
Kent's seam attack of Parnell, Claydon and Matt Coles (3-81) took all 10 of Sussex's wickets and looked to have them well on top when Coles had captain Luke Wright (29) caught behind just after lunch.
But Brown's aggressive innings, which contained 12 fours and lasted 116 balls, combined well with Archer, who continued an impressive match after taking 7-67 with the ball on the first day.
Their stand put Sussex on near level terms at the halfway point, but despite losing Bell-Drummond and Denly to Shahzad, Kent ended the day 95 runs ahead with eight wickets remaining.
Ben Brown and Jofra Archer put on 91 for the final wicket to help Sussex stay in the match against Kent.