Eastbourne batsman Orr hit 119, including 19 fours, as Sussex advanced to 277 for three, but they were stifled for long periods by the hosts, who trail by 293 going into the final day.
Orr’s opening partner Tom Haines was out for 94 just before stumps to Matt Quinn, who took two for 88.
Earlier Jamie Atkins took career-best figures of 5-51 as Kent were dismissed for 165, Hamid Qadri finishing on 30 not out, his highest first-class score.
Kent began day three on 122 for seven and lost Bailey Wightman in the first over when he was bowled by Atkins, but Quinn hung around for 38 minutes, providing useful support to Qadri, before he was caught behind off Ollie Robinson for 11.
By the time Jas Singh was lbw to Hunt for two, Sussex’s lead had been chopped to just 16, but Orr and Haines steered the visitors to 53 for nought at lunch and batted through the afternoon session. Kent never really looked like taking a wicket, but they were tight and disciplined, slowing the run rate to under three and limiting Sussex to 147 for no wicket at tea.
Orr reached three figures when he glanced Harry Podmore to third man for four and was finally out when he cut Singh to backward point and was caught by Dan Lincoln, having faced 254 balls over nearly five hours.
Harrison Ward made six when he drove Quinn to Heino Kuhn, who took a two-handed catch at point and the scoring rate actually slowed to 2.8 during the evening session, only nudging back over three when Travis Head hit sixes from successive balls in the 90th over, followed by two fours. Quinn then had Haines lbw in the third last over of the day, ending a painstaking innings of 233 balls and 354 minutes.
Head and Oliver Carter were unbeaten on 29 and 15 respectively at stumps, but Sussex face a potentially awkward decision over the timing of any declaration on day four.
Orr said: “It feels amazing, it’s almost like a weight off my shoulders. I was very, very nervous when it came round but I’m just so happy and hopefully there’s a few more in there. At first I was just trying to get bat but when I got a bit closer I thought ‘Oh God, it’s starting to happen again.'
"I thought back to Glamorgan and how close I and I was just absolutely focused that I was going to get there, so it was a relief, but obviously joy as well. I spoke to the coaches at tea and I think I was 86 not out or something and I said look I’m a bit nervous (because of) what happened at Glamorgan and they said if it helps just look at how many balls you were facing instead.
"So I just set myself little targets of how many balls I was facing until looking at my score and then luckily got a couple away and got there a little quicker.”