Cook also threw a TD pass to Trevon Pendleton, and Jeremy Langford rushed for 84 yards and a score for the Big Ten champion Spartans (13-1), who finished their season with 10 straight wins and their first Rose Bowl victory since 1988.
Michigan State overcame its first double-digit deficit of the entire season along the way, and the Spartans' FBS-best defense capped a dominant season with one more old-school, smash-mouth performance befitting the centennial celebration of the Granddaddy of Them All.
"It's a special time for all Spartans, and we came here in force," coach Mark Dantonio said. "I'm very happy for our football team, the resilience we showed all season long."
Michigan State's defense yielded just 159 yards in the final three quarters and ended it by stopping the Pac-12 champion Cardinal (11-3) on fourth-and-1 near midfield with 1:46 to play, stuffing a run play up the middle.
Elsworth, a fill-in starter for suspended senior linebacker Max Bullough, hurdled the pile to deliver an electrifying, head-on hit to fullback Ryan Hewitt while his teammates helped out below.
"When I saw their offensive linemen's stance, I knew the way to make a play was to go over the top," said Elsworth, selected the game's defensive MVP.
The huge Michigan State contingent in the Rose Bowl stands roared at the play, and even the stone-faced Dantonio visibly celebrated.
"I get a little excited at the Rose Bowl," Dantonio deadpanned.
The Spartans have long labored behind Michigan, Ohio State and even Wisconsin among the Midwest's top programs, but Dantonio's seven-year rebuilding project in East Lansing has put them on top of the Midwest this season with a perfect run through conference play.
After knocking off the unbeaten Buckeyes in the league title game, Michigan State earned the Big Ten's second Rose Bowl win since 2000.
Tyler Gaffney ran for 91 yards and an early TD for Stanford, and linebacker Kevin Anderson returned an interception 40 yards for a score late in the first half. But the Cardinal couldn't follow up last season's success in Pasadena with back-to-back Rose Bowl wins, managing just three points from their offense after the first quarter.
And Gaffney could only watch as Hewitt was stopped on Stanford's final play.
"You have to give it to Michigan State for stuffing that," said Gaffney, who managed just 24 yards after the first quarter. "Everybody in the building knew exactly what was coming. A run was coming up the middle, and it was a test of wills, and they got the better of us."
Cook led in his own inimitable fashion, making incredible plays and huge mistakes along the way.
No. 4 Michigan St. 24,No. 5 Stanford 20
Stanford 10 7 0 3 20
Michigan St. 0 14 3 7 24
Stan • Gaffney 16 run (Williamson kick), 11:16.
Stan • FG Williamson 34, 1:40.
MSU • Langford 2 run (Geiger kick), 10:45.
Stan • K.Anderson 40 interception return (Williamson kick), 2:07.
MSU • Pendleton 2 pass from Cook (Geiger kick), :28.
MSU • FG Geiger 31, 12:56.
MSU • Lippett 25 pass from Cook (Geiger kick), 13:22.
Stan • FG Williamson 39, 4:15.
Attendance • 95,173.
First downs 11 21
Rushes-yards 36-162 35-65
Passing 143 332
Comp-Att-Int 10-18-1 22-36-1
Return Yards 44 21
Punts-Avg. 5-49.8 6-45.5
Fumbles-Lost 2-0 1-1
Penalties-yards 8-55 4-35
Time of Possession 28:49 31:11
Rushing • Stanford, Gaffney 24-91, Hogan 8-41, Rector 1-27, Seale 1-4, Hewitt 1-0, Team 1-(minus 1). Michigan St., Langford 23-84, Kings Jr. 2-7, Fowler 1-(minus 5), Team 3-(minus 6), Cook 6-(minus 15).
Passing • Stanford, Hogan 10-18-1-143. Michigan St., Cook 22-36-1-332.
Receiving • Stanford, Montgomery 3-21, Rector 2-44, Cajuste 1-51, Hewitt 1-11, Sanders 1-11, Wilkerson 1-7, Gaffney 1-(minus 2). Michigan St., Lippett 5-94, Langford 5-17, Kings Jr. 4-52, Fowler 2-97, Pendleton 2-21, Mumphery 1-20, Gleichert 1-17, Price 1-9, Lyles 1-5.