Murray, however, has not been ruled out for this week's game against the Indianapolis Colts.
Sources told ESPN's Ed Werder that the Cowboys are confident Murray will be able to play against the Colts as they consider this injury to be easily tolerated.
"The biggest question we have to ask ourselves is, is he functional to do his job?" Garrett said. "Can he hold the football? Can he carry it under duress? Can he block? Can he do the things necessary to play the position?
"He's as strong-willed and as determined an individual as I've ever been around, and if anybody has a chance to come back, he does."
Murray suffered a broken fourth metacarpal bone, a source told ESPN.com. The injury occurred late in the fourth quarter of the Cowboys' 38-27 win against the Philadelphia Eagles.
After the game, Murray said he was "good," but he did not elaborate about his hand. He had X-rays taken after the game and was examined further Monday.
The fourth metacarpal is the long bone in the hand that runs from the base of the fourth finger to the wrist. The Cowboys know the bone won't fully heal for six weeks. In terms of risk, there's no difference between playing this week or sitting out one game.
"We have a lot of confidence in those guys," Garrett said. "Obviously we like to give them opportunities in games to spell DeMarco and just to give those guys a shot. I thought last night was a good example of how those guys consistently take advantage of those opportunities. Joseph got a couple of carries and was productive with his. I felt Lance did a really nice job in that game, at the end of the game handling the football and just doing some good things and making some good runs. They weren't splashy runs but they were real positive runs in critical moments of that ballgame."
Should Murray miss the game, it would be a serious blow to an offense that has relied heavily on the NFL's leading rusher. Murray has established career highs in carries (351) and yards (1,687). He needs 87 yards in the final two games to eclipse Emmitt Smith's single-season team record of 1,773 yards set in 1995. He has an outside shot at 2,000 yards, as well, but those chances took a hit Sunday when he was held to 81 yards on 31 carries.
Murray has 11 100-yard games this season and has scored a career-high 11 touchdowns. Randle has 237 yards on 34 carries and has two touchdowns. Dunbar has 91 yards on 26 carries.
Others have dealt with the same or similar injuries as Murray's. Smith missed one game in 1999 after undergoing the surgery following a 13-carry, 140-yard and two-touchdown game against the Minnesota Vikings. He had two straight 100-yard games after his return. In 2006, Terrell Owens suffered a similar injury while with the Cowboys. He had a plate and screws implanted into his hand and did not miss a game, thanks to the Cowboys' bye week. He wore a glove with extra padding on the top of his hand for added protection. In 2011, Denver Broncos running back Willis McGahee missed one game with the injury.
Because Murray carries the ball predominantly in his right hand, it might not be much of an issue, but he has lost five fumbles on the season. He missed 11 games in his first three seasons with ankle, foot and knee injuries.
"When they say the definition of healing, I'm not so sure how much it can heal between now and Sunday," Cowboys owner and general manager Jerry Jones said on 105.3 The Fan in Dallas on Tuesday. "I haven't been shown examples of where someone has had an injury like this. It's not uncommon. I'm not saying it's an everyday occurrence, but it's not uncommon for a back to have a hand injury. They do take some hits on that hand. But I know it's in a place where it can really be protected. We do that and we see that often with different kind of hand injuries. Certainly he's got a lot of parts to his game that work - protection, what he does, frankly, with that hand when he's running with the ball, he usually carries it in his right hand. All of those things, it's conceivable to me, it's not unrealistic that it could work with him being ready."
Information from ESPN's Ed Werder was used in this report.