The Tigers (24-8) made seven of their final eight free throws over the last 1:41 and 13 of their final 16 overall to shut the door on Duke (27-5) and pull off the big win and end a 22-game losing streak to the Blue Devils.
Hammonds, who entered the game shooting 45 percent from the line, drained four straight free throws in the final 43.4 seconds, while Mays, who entered the game shooting 56 percent, made all six of his attempts.
It's the most free throws he's made in a game this season and ties for the most made in a single game in his career. The four three throws are also the most made by Hammonds in a game this season.
"We've been talking about it all year," Hammonds said. "Free throws are going to be big for us in a big-time situation.
"We've been working on it each and every day and we just knew that if we stepped to the line with confidence, they're going to drop. That's the way I went to the line and (James) Mays went to the line."
By winning, the Tigers will now face North Carolina (31-2), the No. 1 seed in the tournament and the No. 1 team in the nation, at 1 p.m., Sunday.
But Clemson is going to savor one of the biggest wins in the history of the program just a little bit before getting ready for the Tar Heels.
"We'll tell them to enjoy this because we do want them to have a good time and I thought they had fun out there today," Tigers coach Oliver Purnell said. "But when we wake up tomorrow, what happened today doesn't matter. We played a good game against Boston College and that didn't matter today. We played well against Duke today and that won't matter tomorrow."
Speaking of free throws, the Tigers won the game despite Duke dropping 16-of-17 from the line, which equates to 94.1 percent, the highest percentage the Blue Devils have ever shot in an ACC Tournament game, and they've played in 120 of them.
But it just wasn't free throws for Clemson that won it. It was the staples of this team and that's pressure defense, 3-point shooting and an inside game.
The Tigers forced Duke into 16 turnovers and a 43.3 percent shooting percentage. Star freshman Kyle Singler was held to just two points on 1-of-9 shooting.
"The turnovers definitely came at the end of the play, and Clemson did a great job of following the play and back-tapping from behind," said Duke senior guard DeMarcus Nelson, who had 14 points and four turnovers. "That is a team defense that we normally don't play against and it is definitely and adjustment we had to make."
Also hurting the Blue Devils was their shooting from beyond the arc. As a team, they went 6-of-26. But there's a reason for it and it's more than just having an off-game.
"They did a good job the last few minutes on switching the ball screens, using their length to press up and make us put the ball on the floor," Duke junior guard Greg Paulus said. "If we were to take a three, it was a contested, tough shot. They did a good job of that."
In the post, Mays and sophomore Trevor Booker were able to do most of whatever they wanted to as they went a combined 14-of-18 from the field for 34 points.
Booker finished with 18 points as missed only two shots, while also grabbing seven rebounds, six of which were offensive. Mays had 16 points and four rebounds.
"I felt (we couldn't be stopped)," Booker said. "I was feeling it. Me and James feed off of each other."
As a result, the team, program, school and fans all get to once again taste the spoils of victory, which hasn't happened too often at the tournament.
"We belong here," Hammonds said. "And that's the way we came into this tournament. We belong here and this is our time."