Nick Begich is one of only five people to win election to the House or Senate posthumously. Three others died before Election Day, but were replaced on the ballot.
1962. Rep. Clement Miller (Calif.). Miller was killed in a plane crash in October, while seeking reelection for a third term. He won by 1.6 percentage points. Donald Clausen was chosen to fill his seat.
1972. Reps. Nick Begich (Alaska) and Hale Boggs (La.). Boggs, then the Democratic House Majority Leader, was campaigning with Begich in a small plane in Alaska when it vanished. Three weeks later, Election Day arrived, and both men -- neither of whom had been declared dead -- won reelection. (Begich won by 12 points; Boggs was unopposed.) Boggs was replaced by his wife Lindy, who served until 1991. Begich was replaced by Rep. Don Young, who won a special election the next March -- after losing to the missing Begich. Young still holds the seat.
2000. Gov. Mel Carnahan. In one of the most famous examples of a politician winning a race despite being dead (also due to a plane crash), Mel Carnahan narrowly defeated former Missouri Gov. John Ashcroft in the 2000 Senate race in the state. Carnahan's widow, Jean, was appointed to his seat but lost a special election in 2002.
2002. Rep. Patsy Mink. Mink won her 2002 House reelection campaign (handily) despite having died weeks earlier from pneumonia. Rep. Ed Case won a special election to replace her.
For each of the past three election cycles, state senators have died and then won reelection to their seats. In 2008, Pennsylvania state Sen. James Rhoades won reelection despite having died in a car crash. In 2010, Jenny Oropeza was elected to the state senate in California after her death from cancer. In 2012, Texas state Sen. Mario Gallegos won election after dying from liver disease.
Three U.S. Senate candidates died prior to Election Day but were replaced on the ballot before voters went to the polls. The most famous is certainly Minnesota Sen. Paul Wellstone who died (in yet another plane crash) shortly before the 2002 elections. Former vice president Walter Mondale was chosen to replace him; Mondale lost by two points. In 1976 and 1978, two other candidates died -- both in plane crashes -- and were replaced on the ballot; one replacement candidate won, one lost.
There don't seem to be examples of people who have died, been on the ballot themselves, and lost. In other words: Dead people appear to be batting 1.000 in their Congressional races. Nonetheless, we do not recommend it as a campaign strategy.