The win kept Michigan State's bowl hopes alive and introduced yet another potential lead running back in Leggett, a sophomore who had one carry and zero yards coming into the game.
On his first rush Saturday - on a third-and-1 less than 90 seconds into the game - Leggett took one step toward a crowded line, bounced it outside and was gone, sprinting 71 yards down the left sideline for a score.
He added short TD runs in the first and second quarters. Sandwiched in between, Cousins connected with Garrett Celek and Keshawn Martin on 13- and 48-yard scoring strikes.
At the end of the first half, Michigan State led 35-0, had forced seven Western punts and outgained the Broncos 358-36.
The Spartans (5-5) hadn't played a non-conference opponent in November since 1966 against Notre Dame. The win snapped a two-game losing skid and gave Michigan State some potential momentum heading into the season's final two games - at Purdue next week and home against No. 11 Penn State on Nov. 21.
Western (4-6) didn't do itself any favors, getting called three times for roughing the passer in the first quarter and dropping more than a few passes.
One bright spot for the Broncos was Brandon West, who set the Football Bowl Subdivision record for career kickoff return yardage, passing SMU's Jessie Henderson. The senior opened the second half with an 80-yard touchdown run through a gaping hole in the middle of the Michigan State defensive line.
Michigan State kicker Brett Swenson also was a record-setter, becoming the school's all-time leader in extra points made.
The Spartans have used a number of running backs this season, using a back-by-committee approach to fill the sizable void left by the departure of Javon Ringer, who moved on to the NFL.
Thanks to injuries and dismissals, Leggett and freshman Edwin Baker started out as the featured backs on Saturday, and Leggett took advantage, hitting the century mark by halftime.
Leggett added a 13-yard TD run in the fourth quarter to make it 42-14.
He was the first Spartan to run for 100 yards in a game since Ringer did it last year.
Fellow sophomore QB Keith Nichol took some snaps, but Cousins got the bulk of the work, completing 22 of 25 passes and no interceptions. Nichol threw a 7-yard scoring pass to Dion Sims with about six minutes to play in the game.
For Western, it was a forgettable performance from a program accustomed to playing tough against the big boys. The Broncos had beaten a team from a Bowl Championship Series conference in each of the previous three seasons.
Senior Tim Hiller, Western's career leader in TD passes and completions, had an off day, going 15 of 37 for a season-low 111 yards. He wasn't intercepted, though, and threw a 12-yard touchdown pass to Blake Hammond on the first play of the fourth quarter.
Michigan State avoided losing at home to a second in-state Mid-American school this season. Central Michigan beat the Spartans 29-27 on Sept. 12.