Cope excelled at baseball in high school, playing catcher at Bethel High School before attending Whitman College, where he was scouted by Major League Baseball teams. A knee injury ended his playing career and redirected him to motorsport full-time. He worked through the short-track ranks in the Northwest before making his Winston Cup debut at Riverside International Raceway in 1982, finishing 36th after an oil leak in his No. 95 car. He ran part-time in the series through the decade, recording four top-ten finishes in 1989 for car owner Bob Whitcomb in the No. 10 Purolator Pontiac and later Chevrolet.
Cope's defining moment came on the final lap of the 1990 Daytona 500. Dale Earnhardt, who had dominated the race from the front, ran over debris in turn three and cut a tire. Cope, narrowly avoiding the resulting incident, drove underneath the slowing Earnhardt, assumed the lead, and crossed the line to win. It was not only his first career victory but his first career top-five finish — his previous best had been a sixth. The win made Cope an overnight celebrity; he appeared on Late Night with David Letterman that week. Later the same season at Dover, he collected a second Cup victory after a fuel strategy played out in his favor.
The promise of 1990 proved difficult to sustain. Cope drove for Cale Yarborough's team beginning in 1993, changing car numbers mid-season from the No. 66 to the No. 98 to reflect a Bojangles sponsorship. During 1994 he won his only Busch Series race at New Hampshire International Speedway. His most consistent season came in 1995 with the Bobby Allison Motorsports team, recording eight top-ten finishes and finishing fifteenth in points, with a runner-up result at Phoenix where he led 34 laps before finishing behind Ricky Rudd.
After Allison's team closed at the end of 1996, Cope moved through a series of rides including MB2 Motorsports and Bahari Racing, where he earned his only career Cup pole at Lowe's Motor Speedway despite struggling for results. He failed to qualify for ten races with Bahari in 1999 and was released.
Cope announced the formation of Quest Motor Racing in 2001 as co-owner alongside drag racer Warren Johnson, though the team failed to qualify for any Cup race that year. Through the 2000s he fielded cars in various NASCAR series under the StarCom Racing banner while also driving. He returned to the Cup Series in 2017 under Premium Motorsports, qualifying for the Folds of Honor QuikTrip 500 at Atlanta — his first Cup start since 2009 — and made additional starts through StarCom Racing.
Cope's most recent Cup start came in the 2021 Daytona 500, driving the No. 15 for Rick Ware Racing in alliance with StarCom Racing. When StarCom shut down at the end of 2021, Cope transitioned away from team ownership and later joined Nick Tucker's Nitro Motorsports in the managing department of a Trans-Am 2 team. His career spanned more than 40 years of active involvement in NASCAR at various levels, a testament to his persistence in an industry that rarely rewards longevity so broadly.
Cope's nieces Angela Ruch and Amber Cope and his cousin Nick Tucker have also raced in NASCAR. His first cousin is NASCAR crew chief Ernie Cope. He has been married twice; his current wife Elyshia served as marketing director at StarCom Racing.