Worcestershire old boy Davies was caught at mid-off and then Hamilton-Brown was well taken by James Cameron at deep extra cover.
Finally, from the last ball of the 13th over, De Grooth's drive off Collingwood caught a leading edge and looped invitingly to extra cover.
Ponting holed out to De Villiers at deep backward square leg, while Marsh was superbly caught by Kallis at extra cover off JP Duminy's occasional off-spin.
His first scoring shot was a six over extra cover off Tom Smith, and further maximums followed off Pedro Collins, Owais Shah and Smith for a second time.
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But Sussex's last hope of victory effectively went when Michael Thornely, diving to his left at extra cover, put down a difficult chance offered by Murtagh on six.
De Bruyn became the fourth Somerset player to reach fifty in the innings when he hit Sidebottom to the extra cover boundary from the 110th ball he had faced.
But Vince struck the ball very cleanly in his 22 not out, while Lumb reached his 50 with his only six, cleanly struck over extra cover off Will Gidman.
New Zealand were in real trouble and their predicament worsened when, with the run-rate already below six an over, Jacob Oram lofted Shahid Afridi (2-17) to Misbah-ul-Haq at extra cover.
Johnson hit off-spinner Suraj Randiv for a four and a six to give them a glimmer of hope, but was run out following terrific work from Mahela Jayawardene at extra cover.
Michael Clarke was confident enough to drive his first ball wide of mid-off for four but he was not good enough to read Sharma's slower ball and holed out at extra cover.
But partner Salman Butt survived to reach stumps on eight not out and Younis Khan drove the day's penultimate delivery from Jason Gillespie through extra cover for four to finish on seven.
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But Brown had his revenge when Trescothick was caught at extra cover, leaving James Hildreth (22) and Trego (20 not out) to do the bulk of the scoring in the closing overs.
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The much-needed breakthrough came via a well-disguised slower ball from Sharma, when Haddin's forward prod spooned up into the air for the perfectly-placed VVS Laxman at short extra cover to leave the tourists 350-6.
It proved tough going on an unresponsive pitch, however, and it was Panesar who ended an opening stand of 72 when Aaron Finch went up the wicket and looped the ball high to Bell at extra cover.
Another slower ball from Sharma, who finished with 4-77, accounted for debutant White, who had just hit a streaky four through the slips and found himself chipping into the hands of Harbhajan at extra cover.
Reverse-sweeps and paddles were picked off with ease, as were a succession of delightful late cuts off the quicker bowlers, and, against the spinners, drives lofted over extra cover as he backed away to the leg-side.
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The worst dismissal was the run out of Jerome Taylor who played a stroke gently off the back foot to extra cover and set off for a run, only to be sent back by his partner.
Morkel's first two overs went for 24 and Charl Langeveldt fared little better as Kieswetter stepped down the pitch to thump him over extra cover for a mighty six before moving across to sweep from off stump for four.
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And the visitors' chase went from bad to worse when they lost key men Cook, who gave a catch to short extra cover, and Ryan ten Doeschate, run out for 22 after a direct hit by Hildreth, with the score on 33.
Having achieved his run landmark in only his 60th Test, Pietersen gave a catch to extra cover off Shakib's slow left-arm and the pressure increased in the next over when Paul Collingwood was trapped lbw on the crease by Hossain.
Evans struck with his fourth ball of the day to have home captain Alex Gidman also held by Hall at slip and later had Chris Taylor caught by Stephen Peters at extra cover the ball after being hooked for six.
Dale Steyn opened up from the other end, but Kieswetter struck a glorious drive past extra cover and Pietersen got off the mark with a straight drive followed by an exquisitely-timed stroke through the covers as 13 came from the over.
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The placid pitch also caused occasional problems for Pietersen, who would have liked more pace to play his shots, but there were still some strokes to savour, none more so than the emphatic extra cover drive that brought up his century.
With 22 needed from the final two overs Butt struck an exquisite six over extra cover off Mills and added two fours in the final over, bowled by Butler, but lost the strike by taking a bye off the penultimate delivery.
Wright thumped O'Brien down the ground for the solitary six of the innings from the final ball of the 16th over but fell two balls later as Rankin returned and the Sussex all-rounder could only spoon off the toe end of the bat to extra cover.
Scott Styris briefly injected some life into the innings in the 15th over with a six over extra cover followed by a pull for four next ball, but Afridi ended the fireworks when a quicker ball was chopped on to his stumps by the New Zealand all-rounder.
An insurer can charge very high risk drivers extra, but not enough extra to cover the higher claims.
Eleven runs short of a century he was dropped at extra-cover - perhaps the first mistake he had made.
On the verge of a maiden double-hundred for his country, Bell picked out short extra-cover with a drive off Smith.
Bravo reached fifty with a stylish straight drive off Siddle and collected another boundary next ball courtesy of an exquisite drive through extra-cover.
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