At tea, West Indies were 199 for four - needing a further 256 to reach their victory target of 455 and square the four-match series at 1-1.
Chanderpaul, their most experienced batsman, was 51 not out after being dropped by Panesar on 18, and Dwayne Bravo unbeaten on 11.
However, left-arm spinner Panesar ended a fourth-wicket stand of 73 when he had Runako Morton lbw for 54.
West Indies, who'd won just one of their previous 27 Tests, resumed after lunch on 99 for three.
Morton was 21 not out and left-hander Chanderpaul, who missed West Indies' record innings and 283-run defeat in the second Test at Headingley with a knee injury, five not out.
Morton, carrying on from where he left off at the interval, confidently on-drove Liam Plunkett for four.
Panesar, who before lunch dismissed Devon Smith, had a couple of vehement appeals for lbw against the gritty Chanderpaul, typically crabbing his way across the crease, rejected by Pakistani umpire Aleem Dar.
With the partnership progressing, England captain Michael Vaughan brought on Kevin Pietersen, rather than himself, to bowl off-spin.
But it was improved fast bowler Stephen Harmison who nearly broke the stand when Chanderpaul, misjudging the bounce, chipped him to mid-on where Panesar, much mocked for his fielding, made ground to the diving chance but couldn't hold on despite getting two hands to the ball.
West Indies were then 127 for three.
Morton, on 40, went down the pitch to Pietersen and his ferocious low straight drive brushed the bowler's fingertips. He went on to his third Test fifty, off 130 balls, with seven fours.
But on 54, after nearly three hours at the crease, Morton padded up to Panesar and Dar gave him out.
Chanderpaul cut and pulled Harmison for succesive fours to complete an 86-ball half-century to follow his first innings 50.
Harmison, wretchedly wayward for most of this series, enjoyed a fine opening spell Sunday of one wicket for 12 runs in eight overs which saw him remove the dangerous Chris Gayle, caught at second slip by Paul Collingwood.
His 500th first-class wicket was applauded by Allan Donald, watching from the pavilion in the former South Africa quick's first match in a five-week stint as England's bowling advisor.
Panesar followed up by removing Smith who on 42 gloved him to Alastair Cook at short leg to end a stand of 53 with Morton.
West Indies, who fielded shoddily, faced a stiff chase after England made 313 in their second innings featuring 106 from Cook - the 22-year-old opener's sixth hundred in his 17th Test.
They would have faced an even bigger target but for seam-bowling all-rounder Darren Sammy's seven for 66 on Test debut.