The home win before a national audience on Thursday marks the team's best start since 2001.
It came amid a shootout between Rams quarterback Jared Goff and Vikings QB Kirk Cousins, who combined for more than 880 passing yards.
Goff's stunning 465 passing yards, with five touchdowns, was his career best. His five passing touchdowns were the most by a Rams quarterback since Kurt Warner in 1999.
"It was a hard-fought night," Goff said. "They're a great team. We had to fight all the way to the very end."
Goff hit Cooper Kupp with two of his four TD throws during a 251-yard first half in which the third-year quarterback flawlessly executed coach Sean McVay's offense.
Todd Gurley then finished with 83 yards rushing and 73 yards receiving while the Rams barely held off the Vikings (1-2-1) and Cousins, who passed for 422 yards and three touchdowns.
The Vikings got the ball back with 2:20 left, but rookie John Franklin-Myers stripped Cousins of the ball near midfield. The Rams recovered and ran out the clock.
The Associated Press contributed to this report.