The confrontation -- as authorities continued to search for the other escapee, David Sweat, Friday -- caps a 20-day manhunt involving more than 1,100 law enforcement officials.
Here is a timeline of the case so far, which has been likened to the blockbuster movie "Shawshank Redemption."
Before the Escape
Mitchell also allegedly confessed to having sex with Matt, Clinton County District Attorney Andrew Wylie told ABC News.
She is accused of hiding hacksaw blades and other tools in frozen hamburger meat, which Corrections Officer Gene Palmer allegedly took to Matt and Sweat. Palmer said in a police statement he did not know what was in the meat and denies knowing the men were planning an escape.
Palmer allegedly gave the escapees art supplies, tools and access to the catwalks behind the walls of their cell to do electrical work in exchange for artwork, according to court documents.
Matt was an avid artist, known for his portraits of celebrities.
Authorities said the escape was being planned for five weeks. According to prosecutors, Mitchell smuggled boxing gloves and lighted glasses, along with blades, chisels and cutting tools into the facility.
Palmer is awaiting arraignment and Mitchell pleaded not guilty.
The Escape: June 6, 2015
The prisoners used power tools to cut through the back of their adjacent cells at the maximum security prison, police said. They broke through a brick wall, cut into a steam pipe, slithered through it and finally emerged outside the prison walls through a manhole, officials said.
On the night of the escape, Mitchell had planned to meet Matt and Sweat at a power plant near the prison and expected to drive them to a wooded area about seven hours away, Wylie said.
He also said Mitchell had talked to the escapees about killing her husband, Lyle Mitchell, who also worked at the prison.
In the end, though, instead of meeting with the men and driving the getaway car, Mitchell complained of a panic attack and chest pain to her husband, who took her to a hospital, Wylie said.
Day 4 of Search: June 10, 2015
Law enforcement officers searched on foot and via helicopter, looking into all of the escapees' possible getaways including the streets of Dannemora and a broken-down building in Essex.
Day 5 of Search: June 11, 2015
An eight-mile stretch of New York State Route 374 near Boquet was closed after bloodhounds picked up the escaped prisoners' scents, officials said. The Saranac Central School District was also closed."
Day 6 of Search: June 12, 2015
Mitchell was arrested and charged with a felony count of promoting prison contraband and a misdemeanor count of criminal facilitation for allegedly helping Sweat and Matt escape from the maximum security facility on June 6. She entered a not guilty plea.
At this point, there still had been no confirmed sightings of the escaped inmates and no conclusive evidence that either of the convicted killers left the area.
Day 9 of Search: June 15,2015
Wearing handcuffs and a black-and-white striped prison uniform, Mitchell made a court appearance where she pleaded not guilty to the charges against her.
Day 10 of Search: June 16, 2015
Joyce Mitchell's husband, Lyle Mitchell, was seen publicly for the first time since his wife was arrested. He was photographed leaving his wife's attorney's office, where he refused to answer any questions at the time.
Three officials briefed on the ongoing search in upstate New York told ABC News that the consensus was that the trail has grown cold and that senior state and federal officials were trying to decide what the next steps should be. ABC News was told by an official that Matt and Sweat kept notes on their cell walls and inside the pipes that they used as their escape route.
Day 12 of Search: June 18, 2015
State police said they were expanding the search area, and that more than 600 officers involved in the search cleared more than 160 unoccupied buildings and seasonal homes in upstate New York. They also searched 585 miles of railroad beds and trails, they said.
Day 13 of Search: June 19, 2015
Palmer, a veteran corrections officer, was placed on administrative leave as part of the investigation, and officers said they were focused on a ground search in the heavily wooded region around the prison.
Day 14 of Search: June 20, 2015
More than 300 officers searched the towns of Amity and Friendship after a witness spotted two men on a railroad line along Route 20 in Friendship, New York, near the Pennsylvania border, police said.
State police had been searching a construction site in Erwin, where men fitting their description were seen in that town walking near the Gang Mills rail yard and seen the next day in Lindley heading for the Pennsylvania border.
Day 15 of Search: June 21, 2015
As officers continued the manhunt for the men in Allegany County, Sen. Charles Schumer (D-N.Y), said Matt and Sweat were possibly "riding the rails."
Day 16 of Search: June 22, 2015
Authorities collected several items from a hunting cabin in Owls Head, New York, including prison-issue underwear, according to officials. The cabin was 30 miles west of the prison from which Matt and Sweat escaped and was reportedly used by several of the prison's corrections officers.
Positive DNA samples from food in the cabin were confirmed to be from both suspects, officials added.
Day 18 of Search: June 24, 2015
Investigators looked into whether Joyce Mitchell may have had sex with inmates, multiple sources told ABC News. Meanwhile, officers continued a search in the area of Owls Head, New York.
Palmer, who had been on administrative leave was arrested and charged with giving Matt and Sweat art supplies and other contraband in exchange for paintings from Matt and information about the behavior of other inmates.
Palmer denied knowing that the men were going to escape or that anything he allegedly gave them would aid in the escape, according to a statement given to police.
Day 20 of Search: June 26, 2015
Matt was shot and killed by a border patrol SWAT team around 3:45 p.m. in upstate New York, according to officials, who said earlier that the he and Sweat could be headed to Canada.
Matt was spotted after an officer saw him walking in the woods in Elephant's Head, New York, about 20 miles south of the Canadian border and just 50 miles away from Clinton Correctional Facility in Dannemora.
New York State Police said they believed Matt was with David Sweat, earlier, and that they've set up a perimeter in the area around where Matt was shot and killed to try and corner Sweat. There was an active gun battle in the woods where they believe Sweat was cornered.