The moves were announced on Sunday.
Moore bruised his left heel and jammed his left knee running the bases in Saturday's 3-2 loss at Tampa Bay.
Bard was in the starting lineup for Sunday's game against the Rays. Seattle's other catcher, Rob Johnson, has been slowed by back stiffness.
With outfielder Milton Bradley on the restricted list, the Mariners did not need to clear a spot on the 40-man roster for Bard. That allowed Seattle to keep left-hander Erik Bedard on the 15-day disabled list and not move him to the 60-day DL, which would have been the corresponding move without the open roster spot.
Bedard, coming back from left shoulder surgery, is taking part in a throwing program and could rejoin the rotation next month.