The 1978 Daytona 500 , the second running of the event , was the 20th race of the 1978 NASCAR Winston Cup season . The 1978 Daytona 500 , the second round of the event , was the 20th race of the NASCAR Winston Cup season in 1978 . The 1978 Daytona 500 , the 20th round of the event , was the second race of the NASCAR Winston Cup season in 1978 .