The spinner trapped West Indies captain Ramnaresh Sarwan (35), Daren Ganga (49) and Runako Morton lbw as the tourists slumped to 217-5 at tea, 336 behind.
Panesar had bowled Devon Smith just before lunch, while Liam Plunkett bowled the aggressive Chris Gayle (30).
England, who declared on their overnight 553-5, had to do without thigh strain victim Matthew Hoggard.
He suffered that problem towards the end of a morning session in which the visitors produced a defiant batting performance.
Gayle and Ganga coped well with the movement through the air that Hoggard generated and the brisk opening spell delivered by Steve Harmison at around the 90mph mark.
Hoggard struggled with his line, upset by the right-hand/left-hand combination, and Gayle profited to push him down the ground and flick through mid-wicket for boundaries.
The left-hander's growing confidence was illustrated when he crashed Harmison through mid-wicket and long-off for two fours in one over.
It prompted the first change after 12 overs and Plunkett struck gold second ball when a leaden-footed Gayle was beaten by a fuller delivery which knocked back his off-stump to make it 38-1.
Smith then became the aggressor, although he had some luck when he got an inside edge which almost crashed into leg-stump and survived a confident lbw shout against Plunkett.
He came through those scares to thrash a half-volley from the Durham youngster through the covers before cracking Plunkett and Hoggard on the up through the covers for three sumptuous boundaries with lunch approaching.
With Hoggard going off wincing in considerable pain, the home side were running out of ideas until Smith played for turn that was not there and Panesar had his first victim with his first ball.
Sarwan began positively to quickly pick up three fours but he was never really comfortable against the slow left-armer and could have been leg-before soon after the lunch interval.
Life was easier against Harmison, who lost his line alarmingly, and sent a succession of deliveries down the leg-side.
Plunkett's accuracy also suffered when he replaced him, with one ball flying wide of the keeper for four wides, and England were desperate for a lift.
Up stepped Panesar to deceive Sarwan, Ganga and Morton - who saw Paul Collingwood drop a chance high at second slip when Plunkett got a delivery to rear up - in almost identical fashion, with Asad Rauf the umpire on all three occasions.
At the other end, Shivnarine Chanderpaul played a relatively quiet innings, giving little away, while Dwayne Bravo began positively.
But a score of 354 to avoid the follow-on looked a long way off when they went off for tea.
Brief Scores (at tea, D-3):
England Ist Innings 553 for 5 decl Vs West Indies Ist Innings 217 for 5 (Darren Ganga 49; Panesar 4 for 52).