Dombrowski accepted an athletic scholarship to attend the University of Virginia, where he played for coach George Welsh's Virginia Cavaliers football team from 1982 to 1985. He won numerous accolades as one of the finest student-athletes in the history of UVa athletics, including being recognized as a unanimous first-team All-American as a senior in 1985. He was also a two-time first-team All-Atlantic Coast Conference (ACC) selection in 1984 and 1985, the recipient of the ACC's Jacobs Blocking Trophy in 1984 and 1985 as the conference's best offensive lineman, and a first-team Academic All-Conference honoree in 1985. After graduation, he was later honored as a recipient of the NCAA's Today's Top Six Award for his combined athletic ability, academic achievement, leadership characteristics and campus involvement. Dombrowski's No. 73 jersey has been retired by the university.
The New Orleans Saints chose Dombrowski in the first round, sixth pick overall, of the 1986 NFL Draft, and he played for the Saints from 1986 to 1996. After seeing action in only three games as a rookie in 1986, he became a consistent starter for the Saints in his second NFL season in 1987. Dombrowski played left tackle until the middle of the 1989 season, and then played left guard until his retirement in 1996. He ultimately played his entire professional career with the Saints and was named to the franchise's 30th and 35th anniversary teams. During his eleven NFL seasons, he played in 151 regular season games and started 137 of them.