The AFL and the NFL decided to pit their champions against each other in the 'Super Bowl'. For the first two Super Bowls, the NFL had two rounds of playoffs to decide their champion while the AFL (only having 2 divisions with 4 teams in each division) only played 1 game (their championship game) prior to the 'Super Bowl'. Consequently between the end of the regular season and the 'Super Bowl', there was one weekend where the NFL had games and the AFL had no games.
To compensate for this, after Super Bowl II the AFL decided to expand their playoffs to two rounds by offering two wild card positions. This watered down the AFL playoffs because half of their teams made the playoffs; 8 total teams, 2 divisional champs and 2 wild cards.
For Super Bowl IV, the Chiefs made the playoffs as a middle of the road team, I believe they were 7-7, 7-6-1 or something like that. They beat the Jets, then they beat the Raiders and faced the Minnesota Vikings in Super Bowl IV. This is why they were such a huge underdog. Not only did 'experts' still consider the AFL an inferior league, they were going to send a 'middle-of-road' team from their inferior league. Yes, as a Viking fan I know what happen.
After the merger of the two leagues I believe the wild card went obsolete only to resurface later (not 100%).
So according to the NFL record books and such, the first NFL wild card team to win a Super Bowl was the 1980-1981 Oakland Raiders. Because the Kansas City Chiefs did it in a seperate league, the NFL does not recognize the accomplishment.