At lunch on the third day West Indies were 97 for two in reply to England's first innings 553 for five declared, a deficit of 456.
They still needed a further 257 runs to avoid the follow-on target of 354.
Left-arm spinner Panesar clean bowled Devon Smith for 21 after the left-hander misjudged a ball that turned into him to leave West Indies 83 for two after a second wicket stand of 45.
Daren Ganga was 20 not out and West Indies captain Ramnaresh Sarwan unbeaten on 12.
Earlier, first change Liam Plunkett removed the dangerous Chris Gayle for 30 with his second ball, which swung in and clipped the left-hander's off-stump after he'd played down the wrong line.
England stand-in captain Andrew Strauss declared on his side's overnight score before Saturday's play began in the opening match of a four-Test series.
That meant Matt Prior, the first England wicket-keeper to score a century on his Test debut, finished on 126 not out off just 128 balls with 19 boundaries.
Ian Bell was 109 not out having shared an unbroken stand of 190 in 212 balls with Prior.
They were two of four England centurions in the innings, with Alastair Cook (105) and Paul Collingwood (111) also reaching three figures.
England, without injured captain Michael Vaughan (finger) and pace bowling all-rounder Andrew Flintoff (ankle), saw Matthew Hoggard take the new ball after it had been announced that the Yorkshireman's wife, Sarah, had given birth to a son named Ernie.
At the other end Stephen Harmison, the hosts' fastest bowler, sent down his first over for England since a disappointing Ashes tour of Australia was followed by the Durham quick's retirement from one-day internationals.
Harmison, who can be wayward, maintained a generally good line in an opening spell of eight overs for 20 runs.
Gayle - one of the most experience batsmen in a line-up now without retired former captain Brian Lara - showed his class by twice driving Hoggard straight off the backfoot for four.
But having struck five fours, his 39-ball innings ended when he was left perplexed by Plunkett.
Smith then played two fine shots in successive balls when he twice cover-drove Hoggard for four. And when Plunkett overpitched, he too received similar treatment from the 25-year-old.
Hoggard left the field with a left thigh strain after the first ball of his 11th over which was completed by Collingwood.
Prior and Bell's hundreds on Friday meant four England batsmen had scored centuries in the same Test innings for the first time since 1938.
However, much of the West Indies bowling was woeful even allowing for the fact that they hadn't delivered any match overs this tour after two days of their lone warm-up fixture against Somerset were washed out.
They didn't help themselves by twice dropping Collingwood in the 30s.
Prior, who subsequently kept tidily, received plenty of loose deliveries which he dismissed with gusto.
But such was the lack of penetration from a quartet of right-arm seamers, all of whom conceded over 100 runs each, that Sarwan felt obliged to bring himself on.
But his occasional leg-spin produced a succession of long-hopes which Prior thrashed to the boundary.
Brief Score (at lunch, D3):
England 553 for 5 decl (Prior 126 n o, Bell 109 n o, Collingwood 111, Cook 105) Vs West Indies 97 for 2 (Gayle 30, D Smith 21, DGanga 20 n o; Panesar 1 for 4, Plunkett 1 for 28).