Yorkshire had wiped out their first-innings deficit of 71 thanks to an opening stand of 90 between Tom Kohler-Cadmore and Adam Lyth, who made his fifth half-century in six innings. But 20-year-old Carson, who took three for 35 from 20 overs, got Sussex back in contention with three wickets after tea.
He had Kohler-Cadmore (21) stumped before Root (5) got a very feint edge to wicketkeeper Ben Brown as he pushed forward. Carson picked up his third wicket just before the close when Jonny Tattersall (8) was smartly taken at slip by George Garton.
With Ollie Robinson pinning Lyth lbw for 66 and Garton having Harry Brook (17) caught behind, it needed a grafting unbeaten 36 from Gary Ballance to hold things together for Yorkshire, who were 163 for five at stumps, a lead of 92.
They had dominated the first two sessions, even though Sussex didn’t lose their fourth wicket until an hour into the day when Tom Clark (33), who had added 62 with Tom Haines, lost his off stump to Duanne Olivier. From 166 for four Sussex declined to 221 with Olivier and skipper Steve Patterson each picking up four wickets.
In the next over Haines, who was in sight of his second hundred of the season, was bowled for 86, which included nine fours, as he tried to cut Dom Bess. It had taken the England off-spinner 18 overs to claim his first wicket of the summer but Bess bowled tidily enough and with the pitch taking spin he could have a bigger role to play in Sussex’s run chase.
Sussex captain Brown counter-attacked effectively with eight boundaries in his 37 off 30 balls and he was very surprised to be given out lbw to Patterson off a clear inside edge. Robinson lost his off stump to the next ball and after lunch Patterson removed Carson and Garton, courtesy of a stunning one-handed slip catch by Kohler-Cadmore, as he finished with four for 26. Olivier had figures of four for 61.
Lyth and Kohler-Cadmore looked comfortable when they began Yorkshire’s second innings and Sussex were hampered by the absence of left-armer Scott Hunt, who will not bowl again in the match because of a side strain.
Carson said: "Getting Joe Root out is obviously a great feeling. Before the game our coach Ian Salisbury said the great players nick 'em and county players miss 'em and that's what happened. I couldn't really contain my excitement, it was a great feeling.
"We did well after tea. We kept bashing away at the top of the off stump and kept the energy up in the field and wickets have tended to come in clusters in this game. The wicket is quite dry and there's a little bit of spin but it's still a good batting pitch. We should be able to chase on that.
"I know I have the backing to go out there and do my best from Sals (Ian Salisbury). I can't thank him enough, as a coach and human being he's been amazing. He's made a massive impact on my career so far."