Tomlin (3-3) was staked to a 6-0 lead and blanked the Cardinals until a pinch-hit single by Matt Adams produced two runs in the seventh. St. Louis put two runners on base in both the first and second innings against Tomlin, but failed to score on the way to having its brief two-game winning streak halted.
The Indians built a four-run lead through five innings before Damon hit his homer in the seventh off reliever Maikel Cleto.
Tomlin gave up eight hits and walked one before giving way to Joe Smith in the eighth.
Smith allowed two hits in his only inning of work, but struck out the side and Nick Hagadone turned in a perfect ninth in a non-save situation.
Jake Westbrook (4-6) took the loss, allowing three runs on eight hits while striking out seven in six innings.