Panesar took six for 137 as West Indies were bowled out for 394 shortly after lunch on the final day, ending the tourists' hopes of a record run chase.
Shivnarine Chanderpaul (116 not out) scored his first Test hundred for two years but his team fell short of their 455 target. The record chase stays at 418 by West Indies in 2003.
Panesar exploited a wearing pitch that offered him considerable bounce and turn while fast bowler Steve Harmison took four for 95 in the innings and showed a more disciplined line than in recent matches.
England have an unbeatable 2-0 lead in the four-match series and the win meant Michael Vaughan became the country's most successful Test captain, surpassing Peter May's 20 victories.
The West Indies resumed on 301 for five but Panesar settled England's nerves early by having Denesh Ramdin (34) caught by Paul Collingwood at slip to the 15th delivery of the morning.
Ramdin had added 62 with Chanderpaul to build West Indies hopes of the record chase.
Darren Sammy's stand of 37 with Chanderpaul that spanned 14 overs was halted when Panesar took a sharp return catch. The bowler, often ridiculed for his clumsy fielding, seemed stunned that he had held the ball.
Jerome Taylor (11) was dropped on one by Collingwood off Panesar but his watchful 56-minute innings ended just after lunch when he fended a short Harmison ball to short-leg. That was Harmison's 200th Test victim in his 53rd match.
Fidel Edwards departed three balls later when he edged Harmison to Ian Bell in the gully. Corey Collymore was the last man, caught at short-leg by Bell off Panesar.
England, who scored 370 and 313, bowled West Indies out for 229 in their first innings.
Chanderpaul's century was his team's only hundred of the series. He batted for over six hours. West Indies' Sammy enjoyed a successful debut, claiming seven wickets for 66 in England's second innings on Saturday.
The first Test at Lord's was drawn and West Indies suffered their heaviest Test defeat of an innings and 283 runs in the second at Headingley. The fourth and final match is at Durham starting on Friday.