They came at him with dogs, guns, all-terrain vehicles and helicopters — but escaped cop-killer David Sweat continued to outrun a massive manhunt on Saturday.

The 35-year-old prison escapee was on the lam alone after three weeks running with fellow escape artist Richard Matt, 49, his next-cell-neighbor at the Clinton Correctional Facility in upstate Dannemora.

Matt’s taste of freedom was cut short Friday when he was gunned down by feds after refusing to drop his shotgun in a confrontation outside a hunting cabin some 40 miles west of the maximum-security prison.

The elder fugitive was pinpointed after reports of a cabin break-in and a shot fired at a camper on a rural highway a few miles south of the tiny upstate town of Malone.

Sweat, though, was nowhere to be seen, and there has been no sight or sniff of him since. Even the bloodhounds had lost his scent.

Instead, he continued to beat the odds — one man, presumably on foot, outpacing a convergence of more than 1,000 local, state and federal law-enforcement officers.

Homes and stores were put on lockdown, and drivers were subjected to checkpoints within a 22-square mile perimeter of Matt’s slaying.

But by late Saturday, searchers were not even sure that the likely armed Sweat was within the search zone, a law-enforcement source conceded to The Post.

“He may have slipped out of the perimeter,” the source said. “But if he did, he’s still nearby.”

Predicted heavy rain Sunday could be Sweat’s best ally.

“There’s torrential rain coming in, and that’s been something we have been battling on and off for a few weeks now,” Clinton County Sheriff David Favro told the Plattsburgh Press Republican. “We don’t want everything all washed away.”

Giant floodlights were erected for a second night Saturday in hopes of finding the likely armed and desperate Sweat, who had been serving a life sentence for killing a Broome County sheriff’s deputy in 2002.

“Mr. Sweat and his accomplices hit the sheriff’s deputy with the car, got out, and shot him 22 times and then ran over him,” Gov. Cuomo said Friday.

Sweat’s last confirmed location — a week ago — was 30 miles west of the prison at a burglarized hunting cabin in the rural upstate town of Mountain View.

The cabin is a 40-minute hike from the nearest dirt road and owned by a group of correction officers, including a high-ranking guard at the Clinton prison.

The cabin had been filled with guns and ammo, officials revealed, and both men likely swiped them.

DNA from both men was found at the cabin, officials said.

From there, at least Matt spent the next week forging a trail 10 more miles west.

Alone or with Sweat — officials weren’t sure — Matt made it to the far side of the next large body of water, Lake Titus, traveling either through the woods of Titusville Mountain State Forest or along the shoulders of any of the three highways connecting the two areas.

Along the way, break-ins at two cabins in Malone may have provided rest and provisions.

On Friday morning, searchers found a campsite littered with candy wrappers where one or both men may also have rested.

Then, just before 2 p.m. Friday, a gunshot was fired at a camper being towed through Malone.

Checking the camper at a nearby campsite, the driver found a bullethole and called authorities.

A search of the area from which the bullet would have been fired led straight to Matt.

They found him fleeing from yet another burglarized cabin in woods off a dirt road near Route 30.

The cabin remained cordoned off Saturday.

Meanwhile, Malone and neighboring towns remained in a state of panic.

“Today I started crying when I knew they caught one,” said resident Stephanie Bannon, a 55-year-old mother of three. “Because I worry, not only for our community but for all of our law enforcement who are out there right now.”

Her son Justin, 30, is a deputy sheriff in Franklin County and among the searchers.

Said local business owner Michael Hurteau, 52, “People are hoping they can catch the second one so this thing can stop.”