Leopold and Jones signed two-year deals while Sauer and Finger agreed to one-year contracts.
Leopold, 26, is coming off his fourth NHL season in which he was limited to 15 games due to three separate injuries. He had two goals and three assists in his first year with the Avs after three seasons with Calgary.
"We believe in Jordan Leopold's abilities," said Avs GM Francois Giguere. "He's entering the prime years of his career and we're confident he'll regain the form that he displayed in his first few seasons in the league."
Sauer, 26, played in 48 games last year, recording six assists and averaging over 18 minutes of ice-time.
Finger, 27, played in 22 games for the Avs after his first career NHL callup Feb. 17 and had a goal and five assists.
Jones, 22, scored 18 goals with 26 assists in 33 games during his junior season at Dartmouth, where he was a finalist for the Hobey Baker Award.