Gurley scored his 10th and 11th touchdowns of the season and quarterback Jared Goff completed 14 of 28 passes for 201 yards for the Rams, who handed the Broncos their fourth consecutive defeat.
"Everybody did their job of opening holes and I took care of the rest," said Gurley. "It is 6-0 that's saying a lot about this team."
Stephen Gostkowski kicked a field goal as time expired and Brady threw for 340 yards as the Patriots handed the Chiefs their first loss with a 43-40 win at Gillette Stadium.
"I am glad we had our last shot and glad we took advantage of it," said Brady.
The Broncos-Rams game was played under chilly conditions of -3 Celsius (25 Fahrenheit) at Mile High stadium after an overnight snowstorm moved through the Denver area.
Gurley's first touchdown was a 10-yard run that gave the Rams a 13-3 halftime lead. His second was a one-yard scamper that made it 20-3.
Rams receiver Cooper Kupp was carted off the field after injuring his left knee in the second quarter, but returned in the second half. He will be further evaluated by team doctors on Monday.
The Rams close out their three-game road trip by travelling to San Francisco to play the struggling 49ers next Sunday.
Elsewhere, Chiefs quarterback Patrick Mahomes threw for 352 yards and four touchdowns to lead a rally and tie the contest late, but the Patriots answered with Gostkowski's field goal as time expired in a wild contest.
Brady threw for one touchdown and eluded a tackle to scamper for another while receiver Rob Gronkowski finished with 97 catching yards for the Patriots. His last catch was clutch and was the 500th of his career.
Brady recorded his 200th career regular season win and also passed former teammate Adam Vinatieri for most career wins in the regular season and playoffs combined with 227.
"We've got a lot of clutch players. I think we have no problem grinding it out," Brady said. "That's what the football season's all about."
New England led 24-9 at the half, but Mahomes went to work in the second half. He passed for four touchdowns and finished 23 of 36 for 352 yards in his first loss as a starting quarterback.
Brown's catch capped a last-minute drive by the Steelers that began on their own 23-yard line with 72 seconds remaining in the fourth quarter.
With the Steelers down 21-20 deep in their own end, Roethlisberger went to work. He first hit JuJu Smith-Schuster for eight yards. He returned to Smith-Schuster for 10 yards, and then connected with him over the middle for another 23 yards.
Brown then hauled in the long pass and ran behind Justin Hunter's block and never slowed down. Pittsburgh added a two-point conversion, when Roethlisberger hit Smith-Schuster in the end zone.
Roethlisberger said he could tell when he lined up for the snap that the Bengals were thinking blitz.
"When you see that look, it's how fast can I get him the ball," said Roethlisberger.
In other games, rookie kicker Jason Sanders made a 47-yard field goal as time expired in overtime to lift the Miami Dolphins to a 31-28 win over the Chicago Bears.
In London, England, Russell Wilson threw for three touchdowns as the Seattle Seahawks rolled to a 27-3 win over the Oakland Raiders in front of 84,900 fans.