The Cowboys (10-4) still aren't guaranteed a playoff spot, but the team is guaranteed an improved record after three straight 8-8 seasons.
Jones' son and the team's executive vice president, Stephen Jones, said the Cowboys have exceeded expectations this season.
"Obviously, it's an improvement this year, and an improvement at a time when a lot of people didn't think we would improve," Stephen Jones said on The Ticket (1310 AM) in Dallas on Wednesday night.
"They actually thought we might go backwards, and here we are at nine wins with three games to go. We've got a lot of work to do. But I really think Jason is the right guy for this job, and I know Jerry thinks the same thing. I certainly see Jason being a big part of our future."
Garrett joined the Cowboys in 2007 as the team's offensive coordinator. He took over the head coaching job in 2010 after the team fired Wade Phillips.
Dallas has not made the playoffs since 2009.
- Will Green