He's been placed on the 15-day injured list in a move that is retroactive to Thursday.
"We got better news than it could have been," New York manager Buck Showalter said. "So we take that as a positive. It could have been worse."
Alonso was hit on his left wrist by a 96 mph fastball from Charlie Morton and left Wednesday's game against the Atlanta Braves. He returned to New York for further testing after initial X-rays revealed no fracture.
Alonso leads the majors with 22 home runs, the National League with 49 RBIs and is hitting .231.
New York had lost six straight heading into a three-game series at Pittsburgh that began Friday.
Mark Canha started at first for the Mets in the opener. Mark Vientos could also be an option, though Showalter said the coaching staff may have to use its "imagination" in thinking of ways to get by without Alonso.
"I'm not going to say someone has to step up and all that stuff," Showalter said. "You've just got to be who you are."
Even with Alonso in the lineup, the Mets have struggled to score consistently. New York is 16th in the majors in runs scored.
In other Mets moves Friday, catcher Tomás Nido has been outrighted to Triple A, infielder Luis Guillorme and lefty Zach Muckenhirn have been recalled from Triple-A and right-hander Stephen Nogosek has been designated for assignment.
The team also said reliever Edwin Uceta had surgery to repair a torn meniscus in his left knee. Uceta initially went on the IL in April with what the team called a sprained left ankle. He is expected to be out for at least an additional eight weeks.
Information from The Associated Press was used in this report.