Chanderpaul, reprieved on 18, was 81 not out after more than four hours at the crease following his first innings 50, while Denesh Ramdin, who was dropped by opposing wicket-keeper Matthew Prior on four, was 26 not out.
Panesar, who took 4 for 50 in the first innings, tried to atone by removing Runako Morton and Dwayne Bravo on his way to figures of three for 87.
England thought they had Dwayne Bravo out for 24 when his edged drive off seamer Liam Plunkett went to the bowler's Durham team-mate Paul Collingwood in the gully, but umpire Billy Bowden, after consulting with fellow offical Aleem Dar, ruled the ball had not carried.
The adventurous Bravo kept going for his shots until, on 49, he was caught off bat and pad by Alastair Cook at short leg after facing 71 balls with eight fours in an 88-run stand with Chanderpaul.
Soon afterwards, with West Indies 253 for 5 in the 82nd over, fast bowler Stephen Harmison took the new ball, and Ramdin then saw Prior just fail to hold a diving chance off left-arm quick Ryan Sidebottom following a leg-glance.
Panesar, who dismissed Devon Smith, had several passionate appeals for LBW against the resolute Chanderpaul, typically shuffling across the crease, rejected by the impressive Dar, who later, rightly, denied several more against Ramdin.
More chances go down
However it was the improved Harmison who nearly broke the stand when Chanderpaul chipped him to mid-on where Panesar, much mocked for his fielding, made ground to the diving chance but dropped it despite getting two hands to the ball.
West Indies, who fielded appallingly, earned themselves a tough chase after England's second innings 313 featuring 106 from Cook, and would have faced an even bigger target but for debutant all-rounder Darren Sammy's excellent return of 7 for 66.
Poor bowling on both sides contributed to a current match extras total of 153, the second highest in Test history behind the 173 shared by West Indies and Pakistan in Barbados in 1977.