Both were restricted free agents who played for Nashville last season.
Saros' three-year deal is worth a total of $4.5 million. Hartman's one-year contract will pay him $875,000.
Saros, 23, posted an 11-5-7 record with a 2.44 goals-against average and .925 save percentage this past season while backing up Vezina Trophy winner Pekka Rinne. He made the NHL's all-rookie team.
The Predators acquired Hartman in a Feb. 26 trade with the Chicago Blackhawks. Hartman had a combined 11 goals and 20 assists in 78 regular-season games with the Blackhawks and Predators in 2017-18.
Hartman, 23, also had two goals and an assist in nine playoff games with the Predators.