Eischen, a left-hander, was injured when he fell to the ground and landed on his glove hand after fielding a ground ball in the seventh inning of a 6-3 win over the New York Mets Sunday night.
A report from mlb.com said Eischen is expected to be out of the lineup 8-12 weeks.
"I landed on my arm and it snapped," the pitcher said.
"We lost a key part to our bullpen," Manager Frank Robinson said. "That's a key blow to us. Now we have to figure out how to plug that gap."
Eischen is 1-0 with a 4.15 ERA this season.