Neshek revealed the horrific tragedy on Twitter, saying, "Please pray for my family. Tonight my wife & I lost our first & only son 23 hours after he was born with no explanation."
Neshek wasn't with the team during the past two days as he was with his wife Stephanee in Florida. She gave birth to Gehrig John on Tuesday night.
"Our hearts go out to Pat and Stephanee and we share in their sorrow of losing their young son," the A's said in a statement. "The entire A's organization will keep the Neshek family in our prayers and thoughts, as they try to come to grips with this tragic occurrence in their lives."
Several of Neshek's teammates and other players around the Majors tweeted their condolences to the right-hander.
"Now is a time to pray for Pat Neshek and his wife. I'm sorry to hear that buddy. Prayers are with you @PatNeshek," A's right fielder Josh Reddick tweeted.
D-backs pitcher Daniel Hudson tweeted, "Just saw a tweet from @PatNeshek prayers and thoughts go out to him and his family. Couldn't even imagine... #BaseballFamily"
The A's clinched the American League West on Wednesday and are set to face the Tigers in the AL Division Series, with Game 1 set for Saturday in Detroit.