1998
The most recent driver to win the last race of the year and the Championship in the same season was Jeff Gordon in 1998. He won the cup and that last race of the year in Atlanta... but he had already clinched before the race started so the pressure was off.
1971
In 1971 Richard Petty did the same thing, winning the last race of the year at Texas World Speedway. Just like Gordon in 1998 he had already clinched the title.
1965
Before that we go back to Ned Jarrett's Championship in 1965. He actually needed a good finish at Moyock in order to win the title, or would have if second place Dick Hutcherson had showed up at the race that day. That was back when many drivers didn't race every race so Ned won the title in '65 but he also ran two more races than any other driver.
1957
Buck Baker won the last race of the year at Greensboro Fairgrounds in North Carolina in 1957. His immediate competitors for the crown also raced but Buck already had clinched the Championship.
1956
In 1956 the final race of the year was in Wilson, NC. Buck Baker claimed the win and the first of two consecutive Championships. Buck had already scored enough points that he didn't need the big win in order to claim the title.
1955
Finally we come to a great performance under pressure. In 1955 both Tim Flock and Buck Baker showed up at Occoneechee Speedway in Hillsboro, NC for the last race of the year. Tim won the race to claim the title despite running three fewer races that year. Buck Baker's third place run wasn't enough.
That is every time in the history of NASCAR's top series that the Champ also won the final race of the season.
Interesting that only once in the history of has a driver been able to win the race while under the extreme pressure of winning a Championship. This is a real testament as to just how much pressure the team is under when it all comes down to the final race.

