After they had conceded a first innings deficit of just 11 runs, Glamorgan seamers Timm van der Gugten and Mick Hogan reduced Sussex to 19 for three when play got underway at 4pm.
Hogan picked up fellow Australian Travis Head in a miserly new-ball spell of 9-4-8-1 while van der Gugten removed Aaron Thomason and Stiaan van Zyl.
But Orr and debutant Oli Carter relieved the pressure in a stand of 43 and Orr then added 49 with skipper Ben Brown (25*) to take Sussex to 111 for four, a lead of 122.
A lengthy mopping-up operation was needed after overnight rain before umpires Ben Debenham and Russell Warren gave the go-ahead. Sussex quickly took the final Glamorgan wicket to give themselves a slender first-innings advantage but their out-of-form top order were soon exposed by Glamorgan’s new ball pair.
Opener Thomason (1) was dismissed by van der Gugten for the second time in the match to a delivery which bounced more than he expected and van Zyl fell to the next ball, trapped on the crease.
Head negotiated the hat-trick ball but was squared up by der Gugten and caught at third slip for five, effectively leaving Sussex 30 for three.
Hogan could easily have picked up two or three more wickets before resting and Orr and Carter were able to play with more freedom, Carter moved to 22 when he offered a tame return catch in off-spinner Andrew Salter’s fourth over, a disappointing end to a promising innings by the 19-year-old debutant.
Dan Douthwaite could have picked up two wickets in his first over. Orr was dropped at second slip and Brown reprieved by wicketkeeper Chris Cooke. Orr made the most of his reprieve by following up the 67 he made on debut against Yorkshire last month with another composed contribution. His 50 came up off 105 balls with seven fours and without him Sussex would have been in dire straits.
Earlier, Glamorgan’s last wicket pair of Salter and Hogan added nine runs before their first innings ended on 206 when Hogan came down the pitch to off-spinner Jack Carson and was bowled for 11. Carson finished with three for 34 and now has 28 Championship wickets. Only Ollie Robinson (29) has more.
Orr said, "I got runs against Yorkshire on my debut which helped massively and I feel in a good head-space at the moment so it's nice to follow it up and hopefully I can build on my score tomorrow.
"The ball did a bit at the start and their opening bowlers bowled really well, but I was able to play a few shots which took the pressure off and being around experienced batsmen like Travis Head and Ben Brown helped a lot. There's a lot of overs tomorrow and hopefully we can get a good lead and put a bit of pressure on them, we definitely feel we still have a chance of winning."