At the close the tourists were 286 for seven in their second innings with teenager Chamara Kapugedera, dropped on 34, a Test-best 50 not out and Chaminda Vaas, who had frustrated England with the bat in the first innings, 24 not out.
England were left needing to beat their previous Trent Bridge record fourth innings victory chase of 284 for six, against New Zealand in 2004, to win this match.
Earlier, Sri Lanka wicket-keeper Kumar Sangakkara had marked his 100th Test innings by scoring 66 - the first fifty of the Test.
Murali, who took three for 62 in England's below-par first innings 229, had also been in fine form in the second Test.
He had claimed all four second innings wickets to fall as the hosts wobbled on the way to a target of 78 that saw them to a six-wicket win at Edgbaston.
It now seems as if only a major innings from an England batsman, on a wearing pitch likely to favour Murali, could deny the 34-year-old Sri Lanka talisman - possibly playing his last Test on English soil - in a match where no-one from the home side had managed a first innings fifty.
"We've got Murali, that's our bonus really," said Sri Lanka fielding coach Trevor Penney.
"The guys were zoned in and they weren't going to give away their wickets cheaply," the former Warwickshire batsman added.
Sri Lanka, 221 for five at tea, lost a wicket soon afterwards when Tillakaratne Dilshan was caught behind off Matthew Hoggard for 32.
New batsman Farveez Maharoof only managed six before he was bowled, playing no shot against left-arm spinner Monty Panesar, who took a Test-best three for 73 from 36 overs.
"We're quite confident we can take three wickets tomorrow and hopefully get get this target," said Panesar. "But it will be difficult with Murali bowling on that wicket."
Paul Collingwood, who top-scored with 48 in England's first innings, had been taken to hospital for an x-ray on a rib injury he sustained while facing paceman Lasith Malinga on Saturday, although the Durham all-rounder later returned to the action after being cleared
Then batsman Kevin Pietersen exited with a sore hamstring, although an England spokesman said he expected him to be fit to bat Sunday while England captain and fast bowling all-rounder Andrew Flintoff, who suffered a recurrence of a left ankle problem, was due to be reassessed before play Sunday.
Away from the injury concerns, Marcus Trescothick - who had been off the field Saturday with a migraine - failed to hold a low slip catch after Kapugedera nicked debutant seamer Jon Lewis when Sri Lanka were 251 for seven.
But the 19-year-old, whose first Test was last month's drawn series opener at Lord's, pressed on to a maiden Test fifty with a single off Liam Plunkett having faced 117 balls with seven fours.
England, with Sri Lanka over 100 ahead at lunch, badly needed wickets.
Not for the first time Flintoff, captain in the absence of Michael Vaughan, provided the breakthrough by having Sangakkara caught by Trescothick, at a wide slip position.
And, six balls later, 143 for four become 148 for five when veteran batsman and former captain Sanath Jayasuriya, playing his first Test since reversing his Test retirement, was lbw on the sweep to Panesar for four.
Fast bowler Plunkett who took the next wicket to fall when Sri Lanka captain Mahela Jayawardene, who had made an untroubled 45, chased a wide delivery and edged it to wicket-keeper Geraint Jones.
Jayawardene knocked over his stumps with his bat and although he raised his hand in apology he could still be fined a portion of his match fee for dissent.
Tharanga, 17 not out overnight, struck Panesar - the first Sikh to play Test cricket for England - for six. But three balls later Panesar had him caught by short leg Alastair Cook for 46.