In what could be the 34-year-old's final Test in England, Sri Lanka left the hosts needing to beat their previous Trent Bridge record successful fourth innings victory chase of 284 for six, against New Zealand in 2004, to win the match and take the three-match series 2-0.
"We've got Murali, who's a different kettle of fish. He's our bonus," said Sri Lanka fielding coach Trevor Penney at stumps on Sunday.
The tourists closed on 286 for seven in their second innings, 288 ahead with two days remaining on a wearing pitch that looks ideal for their talismanic slow bowler.
"I think that wicket here is a nice sort of Sri Lankan wicket," added Penney, having seen Murali take three for 62 in England's below-par first innings total of 229.
Excluding matches against minnows Bangladesh and Zimbabwe, Sri Lanka have not won a Test away from home since beating England at The Oval in 1998.
"It would be very exciting to win," former Warwickshire batsman Penney said.
"We've had a tough eight months, especially in Test cricket. We've been to India and lost that quite badly and Pakistan in similar circumstances. We've been in good positions to win and never quite won.
"It's just enthralling because we've got a lot of young players."
Sri Lanka resume Monday with teenager Chamara Kapugedera a Test-best 50 not out and tail-ender Chaminda Vaas 24 not out.
One downside though was the sight of captain Mahela Jayawardene smashing his stumps after carelessly edging a wide Liam Plunkett delivery to wicket-keeper Geraint Jones to end an otherwise untroubled innings of 45.
Despite raising his hand in apology, Jayawardene could still be fined a portion of his match fee for dissent.
"We just left him alone," Penney said. "He's very disappointed that he as captain played that shot. He's very good natured and normally doesn't do that sort of stuff," the Zimbabwean added.
England spinner Monty Panesar took a Test-best three for 73 on Sunday.
He hailed Muralitharan's ability with the ball but insisted his England colleagues felt they could reach the daunting target.
"It will be difficult with Murali bowling on that wicket," the Northamptonshire left-armer said. "We'd like to bowl them out for under 300.
"Anything 350-plus will be challenging. I think we're pretty confident that anything below 350 we should be fine with that.
"Our recent record of getting a fourth-innings target is very good. The likes of Andrew Strauss, Kevin Pietersen, Marcus Trescothick are players who can chase targets and we're very confident we can do it.
"Whatever target we're set we can chase it down," added Panesar, whose team could, if Sri Lanka's tail wags, find themselves having to surpass England's all-time record Test fourth innings victory total of 332 for seven, achieved against Australia at Melbourne in 1928/29.
Meanwhile England's top order batsmen have several injury worries as they prepare for the challenging task asked of them.
Pietersen suffered a sore hamstring Sunday but was expected to be fit enough to bat, while stand-in captain and big-hitting all-rounder Andrew Flintoff suffered a recurrence of a left ankle problem and was due to be reassessed before play Monday.
Paul Collingwood, who top-scored with 48 in England's first innings, had a rib injury x-rayed Sunday before being cleared to field again after opener Trescothick suffered a migrane Saturday.
England are looking to win their first Test series since beating Australia 2-1 and regaining the Ashes on home soil in September.