Michael Vick completed 13-of-22 passes for 163 yards and a score to help the Falcons (1-0) to their first victory in San Francisco since 1991.
Alge Crumpler caught Vick's TD pass and had six catches for 82 yards. Dunn carried the ball 19 times for 63 yards.
Mora, the 49ers' defensive coordinator the previous five seasons, took over in Atlanta to replace Dan Reeves.
Niners quarterback Tim Rattay played in his first game as the team's full-time starter. After returning from a shoulder injury that sidelined him for nearly two quarters, Rattay threw a crucial interception near the end zone with San Francisco trailing by eight and looking for a touchdown.
Rattay wound up 18-of-31 for 175 yards and a two scores.