West Indies trail the visitors by 212 runs after they bowled South Africa out for 357 on a stop-start day in which there were several interruptions for rain which also resulted in lost play for almost the entire first three sessions of the match.
Kavem Hodge was on 11 not out and will resume on the fourth morning with Jason Holder on 13 as the hosts seek to close the gap on South Africa and deny the latter a sizeable lead that would give them a chance of victory.
The impressive Maharaj bowled 28 overs on the trot from one end for his 3-45 and had two catching chances spilled in the field.
"The ball got a bit wet and doesn't react as much off the surface," Maharaj told reporters. "There is not much off the wicket so I tried to vary the pace and use different trajectories to make life difficult for them."
He bowled opener Mikyle Louis for 35 on the stroke of lunch after a solid opening stand of 53.
Captain Kraigg Brathwaite (35) put on 61 for the second wicket with debutant Keacy Carty (42) before he was needlessly run out by Wiaan Mulder on the ball before tea after failing to slide his bat into the crease while taking a quick single.
After a lengthy delay for rain, South Africa put the brakes on the scoring and Maharaj trapped Carty lbw before Alick Athanaze was caught by Aiden Markram at slip for three.
The home side earlier made light work of claiming the final two wickets in South Africa's first innings after the tourists had resumed on 8-344.
They added 13 runs before being bowled out, leaving Mulder stranded on 41 not out, as left-arm spinner Jomel Warrican took 4-69 and seamer Jayden Seales 3-67.
West Indies have won three of their previous 33 Tests against South Africa, their last victory coming in 2007 and their previous home win 23 years ago.
