Working on the man advantage 8:12 into the game, Kevin Walrod walked out from the right corner in the Minnesota zone and snuck a shot inside the right post past Gopher goaltender Kent Patterson to give Michigan State the lead.
Patterson made two spectacular left-pad saves on Dean Chelios with 7:35 to go, but the Spartans' Brent Darnell pounced on a rebound seconds after the ensuing faceoff to make it 2-0.
Minnesota answered minutes later when Seth Helgeson snapped a shot from the high slot past MSU netminder Drew Palmisano's glove to cut the deficit in half.
Bjugstad scored the first of his two goals 47 later by backhanding a shot from the right hash mark through traffic and in to knot the score, which stayed the same though the end of the second period.
The Gophers got into trouble in the final stanza after playing through a scoreless second period for the third time this season.
With one second left on a Minnesota power play, Walrod tallied his second goal of the night after dueling one-on-one with Gopher defenseman Ben Marshall. Walrod's wrist shot beat Patterson over the left shoulder.
It was the first shorthanded goal Minnesota had given up this season and the first regular-season shorthanded goal allowed by the Gophers since Oct. 22, 2010 against St. Cloud.
MSU capitalized again with the eventual game-winning goal less than two minutes later.
After Erik Haula and Sam Warning came up empty on a two-on-one odd-man rush, senior forward Brett Perlini dropped a pass to freshman Matthew Berry, whose shot from the right circle evaded Patterson.
Bjugstad made it 4-3 with 6:57 left by picking the pocket of Spartan forward Tanner Sorenson, breaking into the MSU zone and zipping a shot through the legs of Palmisano for his second marker of the game.
The goal was Bjugstad's 15th in as many games this year and put him into a tie with Colgate's Austin Smith for the NCAA Division I goal-scoring lead.
The Gophers had numerous chances in the game's waning moments, including a last-minute chance by Kyle Rau, but Palmisano shut the door with an acrobatic save and Michigan State escaped with its sixth win in its last seven games.
The Gophers, who are 6-0-0 in Saturday games in 2011-12, will try to avoid being swept by a non-WCHA team for the first time since Oct. 16 and 17, 1999 when the two teams meet again at 6 p.m on Saturday.