Curran scored a 60 and saw his team home with three balls remaining, having been dropped on 46 by Adam Lyth in a match-defining moment.
Earlier, Lyth himself was dropped on one, before going on to score the fastest fifty in the history of The Hundred, completing the score in just 17 balls.
Exceptional bowling from Sunil Narine helped to restrict the Superchargers to 157 for seven.
Ultimately, Sam Curran's pyrotechnics meant the Superchargers total was not enough.
Jordan Cox, who scored 48 off 29, was scintillating in his supportive role before Tom Curran, with 18 from seven, secured victory for the hosts.
Looking back at the breathless encounter, player of the match Sam Curran told Sky Sports: "Amazing game.
"I think The Hundred almost needed that game that was going to be close.
"We have played in two games at the Kia Oval and it has been amazing.
"The blood is rushing and I am really excited but it is the type of game that was needed and it gives you a lot of confidence, so I am really excited.
"I think the captain Jason had a great day today, the way he moved us as bowlers was so great day today.
"Those are the games you want to play in where they are really close then win as well, so it gives us a lot of confidence."