Devin Lucien turned a short pass into a 40-yard TD with 9:57 to play, and the Bruins (8-2, 5-2 Pac-12) got creative to hold off a lively challenge from the Huskies (6-4, 3-4), who lost starting quarterback Keith Price to a shoulder injury right before halftime.
Jack followed up his 120-yard debut at running back last week by becoming the 13th player in UCLA history to score four touchdowns.
Cyler Miles passed for 149 yards after replacing Price, but the Huskies have lost nine of their last 12 Pac-12 road games.
Damore'ea Stringfellow, Jaydon Mickens and Austin Seferian-Jenkins caught TD passes for Washington, which hasn't beaten UCLA at the Rose Bowl since 1995.
For the second straight week, Bruins offensive coordinator Noel Mazzone dipped into the defense's roster to win an exhilarating meeting of two bowl-bound teams with prolific offenses.
Jack, the Bruins' remarkable freshman from a Seattle suburb, finished with the most rushing TDs since Maurice Jones-Drew set the school record with five scores in 2004. Jack and the hulking Marsh scored the Bruins' first five touchdowns as UCLA stayed right behind division leader Arizona State, which visits Pasadena next weekend.
UCLA coach Jim Mora had been coy about whether Jack would even play offense again, but the freshman jumped in during the Bruins' opening series, rushing for an 8-yard score on his first carry. Used mostly as a short-yardage back, he added two more scores in the first half and finished with 60 yards on 12 carries.
Oh, and he also made five tackles.
Brett Hundley passed for 159 yards for the Bruins, who led 27-7 early in the second quarter and were up 34-24 heading to the fourth. The Huskies kept it close despite losing their starting quarterback on a hit late in the second quarter.
Miles, a redshirt freshman, had played only sparingly this season behind Price, a three-year starter who holds Washington's career record for touchdown passes. Miles went 15 for 22, but threw two interceptions on consecutive throws in the final minutes.
Stringfellow, a freshman with three catches all year, had eight receptions for 147 yards and a score with 8:01 to play. He also had a long TD catch wiped out in the first half by one of Washington's 11 penalties for 113 yards.
Bishop Sankey rushed for 91 yards and a score for the Huskies, who dropped to 7-20 on the road during coach Steve Sarkisian's five seasons.
The Bruins wore their black "L.A. Midnight" uniforms on a rare Friday night game at the Rose Bowl, but the stadium was roughly half-full at kickoff while fans struggled through Los Angeles' usual Friday traffic.
UCLA still led 14-0 less than six minutes in on two short scoring drives set up by Huskies turnovers.
Jordan Zumwalt forced Seferian-Jenkins' fumble on Washington's opening series, and Jack rumbled for UCLA's first score just 2:38 in. Marsh then caught his 2-yard play-action TD pass on a short drive set up by Sankey's fumble near midfield.
After the Huskies made a 95-yard scoring drive capped by Sankey's TD run, Washington could have recovered a fumble by Jack after a 25-yard run - but Shaq Thompson failed to fall on the ball. Two plays later, Jack scored from 1 yard out.
Jack's third TD surge put the Bruins up by 20 points, but the Huskies finally answered with Mickens' short TD catch 6:11 before halftime set up by Malcolm Jones' fumble.
Price left the field grimacing after his final snap of the first half, and he returned to the sideline in the third quarter in a warmup suit and cap.
Miles showed no nerves while leading the Huskies on a 26-yard scoring drive after UCLA fumbled the second-half kickoff, hitting Seferian-Jenkins for a score. But UCLA answered with Jack's fourth TD run, this time from 2 yards out.
UCLA's defense had much more success against Miles, repeatedly stopping the Huskies on third downs. Washington successfully faked a punt and had a field goal attempt blocked later in the third quarter, but couldn't move consistently.
After UCLA stopped the Huskies on fourth down with about 10 minutes left, Lucien turned a short reception into a 40-yard score with a sprint through the secondary, putting the Bruins back up by 17.