Lawrence, 26, is playing next season under the franchise tag that will pay him $17.143 million, and would like to "break the bank" when he hits free agency again next offseason.
"Whatever DeMarcus Lawrence gets, he deserves," said Jones, attending a promotional event at Texas Motor Speedway on Thursday.
"I’ve had a chance to visit with him several times, I’ve had a chance to do some football camps with him, he’s good with kids, and I see why he’s the player he is.
"He has a passion for the game, and he gives you 110 percent every time he steps on the field. So it’s no luck that he’s in the position that he’s in. He’s worked for it."
Lawrence is coming off the best season in his career. He had 14.5 sacks, more than double the second-place sacker on the Cowboys — David Irving, with seven. The 14.5 sacks were tied for second-most in the league.
Most impressive may be that those 14.5 sacks led to a combined loss of 160 yards for opponents. Additionally, he had a team-leading 52 quarterback pressures, more than the next two Cowboys combined – Tyrone Crawford (26) and Maliek Collins (25).
The Cowboys have stated a desire to sign Lawrence to a long-term deal, but neither side came close in talks this offseason.
Jones, a 15-year NFL veteran, finished his career with 57.5 sacks (sacks were not officially recognized by the league until 1982). He reached double-digits twice in his career -- 1985 with 13 and 1987 with 10.