The Cowboys (4-7) broke a three-game losing streak and found themselves relying on their veteran quarterback, who was demoted prior to the game in place of rookie Drew Henson.
Henson, however, could complete only four of 12 passes for 31 yards in the first half and had an interception returned 45 yards for a touchdown by R.W. McQuarters.
After the team's played to a 7-7 halftime tie, Testaverde replaced Henson and drove the Cowboys 61 yards in eight plays for the go-ahead touchdown early in the fourth period.
Dallas took the lead on Testaverde's five-yard pass to Darian Barnes.
The Cowboys scored again four minutes later on a four-yard run by Jones following a Chicago turnover. Jones scored his first NFL touchdown in the opening period on a 33-yard run and picked up 150 yards on 33 carries.
Chicago (4-7) managed just 49 yards on the ground and 91 yards in the air while picking up an average of just 2.5 yards per play.