At stumps on the fourth day West Indies, looking to level the four-match series at 1-1, were 301 for five - needing a further 154 runs to reach their victory target of 455 in a minimum of 90 overs on Monday's final day.
Chanderpaul, reprieved on 18, was 81 not out after more than four hours at the crease following his first innings 50.
Denesh Ramdin, dropped by opposing wicket-keeper Matthew Prior on four, was 26 not out.
Panesar, who took four 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.
It was not the first time experienced left-hander Chanderpaul, now in his 103rd Test, had played a key role in a large run chase.
Back in May 2003 he made 104 when West Indies scored a Test record fourth innings winning total of 418 for seven against Australia in Antigua.
West Indies started Sunday's final session on 199 for four with the 32-year-old Chanderpaul 51 not out and all-rounder Bravo unbeaten on 11.
England thought they had 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 Billy Bowden, after consulting with fellow umpire Aleem Dar, ruled the ball had not carried.
The dashing Bravo kept going for his shots until, on 49, he was caught off bat and pad by Alastair Cook at short leg.
Bravo, who faced 71 balls with eight fours, put on 88 with Chanderpaul.
Soon afterwards, with West Indies 253 for five in the 82nd over, fast bowler Stephen Harmison took the new ball.
Ramdin then saw Prior just fail to hold a diving chance off left-arm quick Ryan Sidebottom following a leg-glance.
West Indies, who'd won just one of their previous 27 Tests, were 99 for three at lunch.
Morton was 21 not out and Chanderpaul, who missed West Indies' record innings and 283-run defeat in the second Test at Headingley with a knee injury, five not out.
Panesar, who dismissed Devon Smith, had several vehement appeals for lbw against the gritty Chanderpaul, typically shuffling across the crease rejected by the impressive Dar, who later, rightly, denied several more against Ramdin.
But 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 were then 127 for three.
Morton, who went to his third Test fifty off 130 balls, with seven fours exited on 54 when, after nearly three hours at the crease, he padded up to Panesar and Dar gave him out.
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 by second slip Collingwood.
West Indies, who fielded shoddily, had a tough chase after England's second innings 313 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 debutant all-rounder Darren Sammy's seven 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.
England 1st Innings 370 (I Bell 97, A Cook 60; C Collymore 3-60, F Edwards 3-94)
West Indies 1st Innings 229 (S Chanderpaul 50; M Panesar 4-50, R Sidebottom 3-48)
England 2nd Innings 313 (A Cook 106, K Pietersen 68; D Sammy 7-66)
West Indies 2nd Innings (overnight: 22-1)
Fall of wickets: 1-4 (Ganga), 2-35 (Gayle), 3-88 (Smith), 4-161 (Morton), 5-249 (Bravo)
To bat: D Sammy, J Taylor, C Collymore, F Edwards
England: Andrew Strauss, Alastair Cook, Michael Vaughan (capt), Kevin Pietersen, Paul Collingwood, Ian Bell, Matthew Prior (wk), Liam Plunkett, Stephen Harmison, Ryan Sidebottom, Monty Panesar
Umpires: Billy Bowden (NZL) and Aleem Dar (PAK)
TV umpire: Peter Hartley (ENG)
Match referee: Alan Hurst (AUS)