Chirinos, McKay, Kittredge and Poche are all out for the season.
"You always expect some injuries, they're going to occur,'' manager Kevin Cash said Sunday. "I think we all knew that they were going to be elevated this season in particular. But no doubt this is a little head scratching.''
Anderson leads the Rays in saves (three) and holds (four). He hasn't allowed an earned run in 11 appearances and has 15 strikeouts and just one walk in 9 1/3 innings pitched.
Despite the injuries, the Rays enter Sunday's games in first place in the American League East.
Fleming started Sunday's game against the Blue Jays and pitched five innings of two-hit ball to get the win as Tampa Bay defeated Toronto 4-2. John Curtiss worked 1 2/3 perfect innings to record his first career save.