England, set a target of 110 to win, survived a couple of scares before finishing on 111 for three after West Indies quick Daren Powell struck either side of tea on the final day to dismiss Alastair Cook and Andrew Strauss.
Michael Vaughan, the England captain, was 48 not out and Paul Collingwood, who'd scored 128 in the first innings five not out after he hit the winning runs on his Durham home ground.
Kevin Pietersen (28) fell shortly before the finish after adding 76 with Vaughan.
Earlier, left-arm spinner Panesar took five for 46, the sixth time he had taken five or more wickets in a Test innings, in West Indies second innings total of 222.
Chanderpaul, who had batted for nearly 18 hours since last being dismissed this series, was the final man out for 70, bowled sweeping against Panesar.
The 32-year-old Guyanese left-hander ended the series with an average of 148.66 but rarely received the support his batting endurance deserved.
At lunch West Indies were 131 for four, a lead of just 18, with Chanderpaul 32 not out and all-rounder Bravo unbeaten on 19.
Bravo, not a naturally defensive player, decided to take the attack to England and when Panesar came on, in the seventh over after lunch, he lofted him over his head for four.
But next ball, trying to drive Panesar down the ground, Bravo holed out to Ryan Sidebottom at mid-off.
Bravo's 43 featured nine boundaries and he put on 68 with Chanderpaul. But West Indies were now only 49 in front with five wickets left.
Their position worsened when Marlon Samuels (two) was undone by a sharply-spinning Panesar delivery and edged to Collingwood at first slip.
Panesar struck again when Denesh Ramdin (four) was bowled by a fine delivery that pitched on middle and hit off-stump.
Chanderpaul soldiered on, completing a 127-ball fifty.
It was his fifth successive fifty of the series after he made 74 in the drawn opener at Lord's, 50 and 116 not out in England's 60-run third Test win at Old Trafford and 136 not out in the first innings of this match.
Significantly, he missed West Indies innings and 283-run defeat, their heaviest innings loss in Test cricket, in the third Test at Headingley with a knee injury.
Durham fast bowler Stephen Harmison then struck twice when Powell, for the second time in the match, irresponsibly holing out before Edwards's off-stump was uprooted.
Harmison bowled at sharp pace after the England and Wales Cricket Board announced Tuesday he would have to undergo surgery for a "non-serious hernia".
West Indies resumed Tuesday on 83 for three, still 30 runs behind.
Chris Gayle was 52 not out and fellow left-hander Chanderpaul unbeaten on 16.
Gayle exited for his overnight score, the opener's best of the series, when he drove loosely outside off-stump against Matthew Hoggard (three for 28) and edged to wicket-keeper Matthew Prior.
England, who'd been in trouble at 165 for six in their first innings, saw all-rounder Collingwood steer them to 400.
West Indies' first innings also featured a recovery after they'd collapsed to 55 for four. Chanderpaul and Bravo (44) added 86 for the fifth wicket in a total of 287.