Oliver comes from a racing family; his father Ray Oliver rode with success until meeting his death in a race fall at Kalgoorlie, Western Australia. The younger of two sons, Damien has reached greater heights than his brother Jason Oliver who was killed in a fall at Belmont Park Racecourse, Western Australia when a young horse he was riding broke both front legs and fell during a race trial. The horse was found to have been administered phenylbutazone prior to the trial and this was thought to be a contributing factor in the accident.
Damien returned to riding in 2006 after serious back injuries sustained in a fall at Moonee Valley, and rode the Japanese horse Pop Rock in the 2006 Melbourne Cup, which finished second to stablemate Delta Blues. In the 2007 Melbourne Cup, he placed second to Efficient on English horse Purple Moon.
Oliver has won the Melbourne Cup on Doriemus (1995) Media Puzzle (2002), the Caulfield Cup on Mannerism (1992), Paris Lane (1994), Doriemus (1995), and Sky Heights (1999), and the Cox Plate on Dane Ripper (1997) and Northerly (2001). He was also the regular rider of Lee Freedman's champion sprinter Schillaci (1991-95) and top filly Alinghi (2003-05). In the 2007 Goldern Slipper, Damien completed the grand slam of Australian racing by winning the 2 year old race on the John Hawkes trained Forensics.
The movie The Cup is a movie on Damien, his relationships with his family and the 2002 Melbourne Cup win on Media Puzzle.