The 49ers received Bibbs and Denver's 2017 fifth-round (177th overall) draft choice for San Francisco's fourth-round draft choice next year.
The 5-foot-11, 203-pound Bibbs signed with the Broncos as an undrafted free agent in May 2014.
After spending the majority of his rookie season on the Broncos' practice squad, Bibbs appeared in 13 games over the last two seasons, recording 29 rushes for 129 yards with two receptions for 75 yards and one touchdown.
Bibbs attended Colorado State, where he played in 14 games and rushed for 1,741 yards and 31 touchdowns in his only season with the school in 2013.
In addition, the 49ers traded up in the fourth round on Saturday to draft Utah running back Joe Williams.