Jason Carter became the first in his family to win elected office since his grandfather took the White House more than three decades ago.
Unofficial results showed Carter claiming 65 percent of the vote to fellow Democrat Tom Stubbs' 23 percent.
The 34-year-old Carter celebrated at a restaurant Tuesday night with his grandparents and other family members.
During the race, Carter focused more on the issues than his famous grandfather. His website made scant mention of the former president, instead focusing on the younger Carter's life as a husband, father, Peace Corps volunteer, attorney and Democratic activist.
Just days before the election, however, Jimmy and Rosalynn Carter stumped with the candidate, passing out fliers in an Atlanta neighborhood and surprising residents going about their usual Saturday business.
The district is heavily Democratic and Jimmy Carter is still a popular figure among some Georgia Democrats. The elder Carter served two terms in the Georgia Senate before becoming the state's governor in 1971. His son - Jason's father Jack - was the Democratic nominee for the U.S. Senate in Nevada in 2006 but lost to Republican John Ensign.
Key issues in Tuesday's race centered on solutions to metro Atlanta's transportation problems and opposition to funding cuts to schools.
The seat became vacant when former Democratic state Sen. David Adelman left to become the U.S. ambassador to Singapore. The district covers portions of Atlanta and Decatur in DeKalb County.
It was one of three open legislative seats up for grabs on Tuesday.
In House District 12 in North Georgia, Republican Rick Jasperse defeated Republican Truett Moss and Democrat Jerry Nally.
Also in North Georgia, Republican Butch Miller claimed state Senate District 49 defeating Republican Jimmy Nortman and Libertarian Brandon Givens.
Both of those districts became vacant when state Rep. Tom Graves and Sen. Lee Hawkins stepped down to run for Congress to replace Nathan Deal. In that race, the two former state lawmakers advanced to a June 8 runoff.