Murray shook off a pair of first-half interceptions, including one returned for a touchdown, and passed for 427 yards - also a Bulldogs' bowl record - against the nation's top-ranked passing defense. He was named the game's most valuable player.
Georgia (12-2) reached 12 wins for the third time in school history.
Nebraska (10-4) lost its third consecutive bowl game, and finished the season with two straight woeful defensive performances. The Cornhuskers lost the Big Ten championship game 70-31.
The Cornhuskers led 24-23 at the half, but committed two of their three turnovers in the final 30 minutes.
Taylor Martinez had two interceptions and two touchdown passes for Nebraska and Rex Burkhead rushed 140 yards in his final college game.
Nebraska's offense finished with 443 total yards, but the Bulldogs defense was stingy when it needed to be.
They sacked Martinez five times, with All-American linebacker Jarvis Jones notching two. Damian Swann had both Georgia interceptions.
Murray found Keith Marshall free near the sideline for a 24-yard touchdown pass that gave Georgia a 38-31 lead with 14:52 to play in the game.
Following a Nebraska punt, Murray then hit Chris Conley for a short pass in space over the middle. Conley outran the Cornhuskers defenders for an 87-yard score.
That play came after a 49-yard touchdown catch by Conley that helped Georgia tie the game in the third quarter.
Nebraska clung to a 24-23 lead the half, taking back the advantage late in the second quarter via a 16-yard strike from Martinez to Burkhead.
After a pair of early turnovers, Murray seemed to be getting Georgia's offense back on track, as the Bulldogs punctuated a four-play, 38-yard drive with a 24-yard touchdown run by Todd Gurley to make it 23-17.
Both Bulldogs' turnovers came via Murray interceptions as the nation's top ranked passing defense crowded Georgia's receivers and disguised their coverage schemes.
Gurley helped open things up, though, rushing for 81 yards in the opening 30 minutes.
Nebraska also allowed Georgia's defense three sacks, which thwarted some promising Cornhuskers drives.
Nebraska's defense didn't need long to force the game's first turnover, picking off a Murray pass on Georgia's opening drive of the game.
The Cornhuskers went three-and-out on their first offensive touch, and were forced to punt.
But that attempt by punter Brett Maher was blocked by the Bulldogs' Shawn Williams. There was a scramble for the ball in the end zone, but it trickled out of the back for a safety.
The Bulldogs then went up 9-0 their ensuing possession following the free kick, capping an 80-yard drive with a 29-yard touchdown pass by Murray to Arthur Lynch.
Nebraska's offense finally got going on its next touch, responding with a 14-yard touchdown pass from Martinez to Jamal Turner.
The Cornhuskers' defense then made with 14-9 when Will Compton came up with their second interception in the half on Murray and returned it 24 yards for the score.
Murray set a new Georgia record for passing yards in a season and touchdown passes with 35. Jones set a school single-season record for sacks in a season at 14.5. The previous record of 14 was held by David Pollack.
Nebraska dropped to 12-7 against SEC opponents in bowl games.