The 29-year-old American carded a 5-under par 65 on moving day at Royal Birkdale.
He sits on 10-under par overall, two clear of New Zealand's Ryan Fox (62) and South Korea's Si Woo Kim (67).
The chasing pack includes Ryan Gerard and Lucas Herbert on 7-under, whilst Bryson DeChambeau sits a further stroke adrift.
Home favourite Tommy Fleetwood will begin his quest for a first major title five back of the lead.
Fleetwood has enjoyed incredible home support in Southport and played his first 11 holes on Saturday in 3-under but stumbled down the stretch, dropping shots at the 15th and 18th.
World number one and defending champion, Scottie Scheffler, is six back on 4-under alongside Xander Schauffele, Jon Rahm and Shane Lowry.
Masters champion Rory McIlroy battled to make the cut on Friday but could only produce a 2-under 68.
At eight strokes adrift of Burns' lead, the Northern Irishman looks like he'll have to wait another 12 months to have a crack at winning his first Open title since 2014.
See how the day unfolded...
L. Herbert -7 (71), Suber -6 (70)
Herbert does well to navigate a two-putt from 50 feet.
But he'll be a tad disappointed with today's performance, particularly given the conditions.
Still, he's three back with 18 holes remaining at The Open.
Suber misses from 12 feet to card a level par 70.
Herbert -7, Suber -6 (18*)
Lucas Herbert finds trouble off the tee and needs to talk to a rules official about taking relief from the barriers down the left side of the fairway.
In the meantime, Jackson Suber plays his approach and hits a beauty to 12 feet.
Herbert eventually hits his shot and finds the green.
Herbert -7, Suber -6 (17)
Lucas Herbert is the latest not to make a birdie at the 17th.
In truth, he's fortunate not to walk off with a bogey after missing the green with his third.
Jackson Suber slips back after leaving himself short-sided with his approach.
Gerard -7 (69)
Ryan Gerard has a monster birdie putt at the 18th.
He watches his golf ball stop a few feet short but tidies up nicely.
DeChambeau -6 (69), Burns -10 (65)
DeChambeau chips to approximately 10 feet but his effort stops in the jaws.
That's a hugely disappointing way to finish.
Sam Burns' birdie attempt from across the green just grazes the right edge.
That's a 5-under 65.
Gerard -7, Young -4 (17)
Ryan Gerard can't take advantage of the par-5 17th but it's another par nonetheless.
DeChambeau -7, Burns -10 (18*)
Bryson does get lucky with his tee shot, but his approach is poor. He misses the green right and will have work to do to save his par.
He gets a great reception as he marches towards his golf ball.
Wallace -2 (72), Kim -8 (67)
Si Woo Kim taps in for par at the 18th to post a 67.
Herbert -7, Suber -7 (16)
Herbert misses the green at the 16th and pays the price.
Suber walks off with a par.
DeChambeau -7, Burns -10 (18*)
Bryson is pumped up.
He takes a wild lash at the ball on the 18th and it immediately goes left and some marshals are forced to duck for cover.
Looks like he's avoided the bunkers, but a very tricky angle.
Sam Burns goes in the opposite direction.
DeChambeau -7, Burns -10 (17)
Boom. DeChambeau takes his time but slots his birdie putt into the dead centre of the cup.
Sam Burns follows him in from five feet for his birdie and is now two clear at the top of the leaderboard.
DeChambeau -6 (17*)
Here comes DeChambeau's approach.
And it's a corker. It flies over the pin and spins back a foot or so. A great look incoming.
Herbert -8, Suber -7 (15)
Lucas Herbert drops back to 8-under after finding trouble off the tee at the par-3 15th. He misses the green left and elects to putt up and over the ridge - but badly misjudges it and leaves himself with far too much to do to save par.
Suber's tee shot finds the putting surface, but rolls off the green and he can't save par.
B. DeChambeau -6 (17*)
A poor tee shot forces Bryson DeChambeau to lay-up at the par-5 17th. He's some ways back but has got a decent angle to the pin. Should give himself a decent look for birdie.
Kim -8 (17)
Si Woo Kim is the latest to take advantage of the par-5 17th, who pitches to gimme range.
B. DeChambeau -6 (16)
Safely aboard in two shots for Bryson.
The trouble is, he's quite far away putting for his birdie.
It's a great effort, but just burns the edge.
He can't believe it.
Will he finish strong?
Scheffler -4 (70)
Scheffler ends his day with another par at the 18th.
That's a level par 70.
He'll need something extra special tomorrow if he is to defend the Claret Jug.
Gerard -7, Young -5 (14)
Superb stuff from Gerard, who has been forced to be very patient today.
After nine pars in a row, he makes back-to-back birdies.
S. Burns -9, B. DeChambeau -6 (15)
Sam Burns birdies the 14th to move one clear of Lucas Herbert.
He finds the middle of the green at the 15th and two-putts for par.
DeChambeau is closer at the 15th and finds his first birdie since the 6th hole to get back to 6-under.
T. Fleetwood -5 (69), J. Rahm -4 (70)
Huge cheers as Tommy Fleetwood saves his bogey to post a 1-under 69.
Rahm produces a moment of magic from the sand to save his par, too, and card a level-par 70.
We'll get some reaction from Fleetwood shortly.
Scheffler -4, Molinari -2 (17)
Scottie Scheffler gives a sarcastic fist pump as another decent attempt at birdie goes begging.
He's made absolutely nothing on the greens today.
Fleetwood -6, Rahm -4 (18*)
Tommy Fleetwood finds one of the many fairway bunkers at the 18th and is forced to punch out.
His approach is goes left and settles just in front of the grandstand.
He elects to putt. It's a decent attempt. About 10 feet or so for bogey.
DeChambeau -5 (14*)
Bryson DeChambeau finds the greenside bunker over at the par-5 14th. He's in real trouble as it looks like he has no stance and no shot.
What to do, what to do.
Herbert -9, Suber -8 (12)
Jackson Suber plays the better tee shot at the par-3 12th, albeit slightly right, but cannot hole the 12-footer for birdie.
Herbert leaves himself and lengthy putt and does well to save his three.
Fleetwood -6, Rahm -4 (17)
Trouble off the tee forces Tommy to lay-up at the par-5 17th.
He plays a nice wedge shot but at 15-feet it was always an outside chance.
Rahm, meanwhile, gets back to 4-under.
