Joe Harris and Malcolm Brogden both had 17 points for Virginia, which became the second No. 1 seed to be eliminated, joining Wichita State.
Adreian Payne had 16 points for Michigan State and his 3-pointer with 1:29 left gave the Spartans the lead for good at 54-51. After a miss by Brogdon, the 6-foot-10 Payne turned point guard, finding Dawson with a line drive lob pass for a dunk with 52 seconds to go.
Harris hit a 3 with 39 seconds left to bring the Cavaliers within 56-54.
Payne found another way to help the Spartans, making two free throws with 32 seconds left for a 58-54 lead.
Brogdon hit a 3 with 1.1 seconds left to bring Virginia within 60-59. Gary Harris made one of two free throws to make it 61-59 and Virginia's last attempt was well off the mark.
Michigan State came up with a defensive effort that Virginia is known for. The Spartans held the Cavaliers to 35.1 percent shooting (20 of 57), well off the 45.7 percent they shot during the season and nowhere near the 54.3 percent they shot in the two NCAA tournament games.
Dawson, who averages 11.0 points per game, has scored in double figures in six straight games and he had a career-high 26 points in the third-round win over Harvard.
Michigan State started the second half by missing seven of its first eight shots and committing five turnovers. Virginia took advantage to turn a 31-27 halftime deficit into a 38-34 lead. The Spartans went on a 13-2 run to go up 49-42 with 6:10 to go.
UConn 81, Iowa State 76 • DeAndre Daniels scored 27 points, 19 in the second half, and UConn held off Iowa State to reach the East Regional final a year after the Huskies were barred from the NCAA tournament.
Daniels hit his first six shots after halftime, the only Husky to make a field goal for over 8½ minutes. His 3-pointer gave seventh-seeded UConn a 49-32 lead.
The Cyclones rallied late, pulling within 67-63 with 2½ minutes remaining. But senior Niels Giffey hit a 3 in the corner for his first points since the game's opening moments, and when the Huskies (29-8) made their free throws in the final minute, the UConn fans packing Madison Square Garden could celebrate.
Dustin Hogue scored a career-high 34 points for third-seeded Iowa State (28-8).