Payne had 16 points and Dawson had 24 points and 10 rebounds as the Spartans fought through a tough Virginia defense to win 61-59 and move to the Elite 8.
Payne had seven of Michigan State's first 10 points then Dawson had eight in a row, but the Spartans went cold from there.
As the Spartans went without a field goal for 6:27, Virginia went on a 12-0 run and took a 25-23 lead.
Virginia went on a 12-0 run and the Spartans went without a field goal for 6:27 as Virginia took a 25-23 lead.
Gary Harris broke the team's scoring drought – and his – with a driving and-one to give MSU the lead back 26-25.
The basket sparked an 8-2 run to close out the first half and lead 31-27.
Dawson had 11 first-half points and five rebounds and he carried it over to the second half, having MSU's first five points of the half.
But the Spartans continue to trail by a basket or two for much of the half, until a 7-0 run keyed by a Payne dunk, a Dawson layup and a Travis Trice 3-pointer put MSU ahead 43-40.
The 7-0 run became a 13-2 run as Michigan State opened it up to 49-42.
Virginia came back to tie it on a 7-0 run, capped by a 3-pointer from Justin Anderson, at 51-51.
Payne hit a 3 to put MSU up 54-51 and followed it with an alley-oop to Dawson.
Virginia's Joe Harris came back with a 3 that rolled around the rim three times before going down to make it 56-54.
Payne hit a pair of free throws, then Virginia point guard London Perrantes hit a pair to make it 58-56. Trice followed with a pair of free throws with 8.6 seconds left.
Virginia's Malcolm Brogdon hit a 3-pointer with 1.4 seconds left.
A foul before the final inbound play put Gary Harris at the line, where he iced the game the first free throw and missed the second intentionally and Anderson's full-court heave fell short.