Mora, who had engaged Tampa Bay starter Scott Kazmir in a long at-bat in the first inning, crumpled to the turf after running out a grounder in the third.
The veteran's injury was classified as a Grade I strain, and he was listed as day-to-day.
"[Head athletic trainer Richie Bancells] told me that it's a minor strain," said Trembley of Mora's injury. "He doesn't believe that he'll be able to play before the end of the series. I was in there talking to the front office and we're going to see what we can do to get somebody in here for tomorrow."
The ailment may have a ripple effect on Baltimore's lineup. Alex Cintron replaced Mora in the lineup, and he will likely start as long as Mora is injured. Alternatively, the Orioles could use designated hitter Aubrey Huff at third base to provide more offense.
Mora, who leads Baltimore in RBIs (97) and is second in home runs (23), entered Friday's game as one of the hottest hitters in baseball. The third baseman had the highest batting average (.399) of any American League regular since the All-Star break, and he also ranked third in home runs (12) and sixth in on-base percentage (.439) since the intermission.
The Orioles, who have been carrying a 13-man pitching staff, had just three reserves on the bench Friday night. All three of the reserves were called on in Tampa Bay's one-sided rout. Baltimore is expected to add reinforcements as early as Monday, when the league allows each team to expand its rosters to 40 players for the final month.
Spencer Fordin is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.