Neshek's son, Gehrig John, died 23 hours after his birth. He posted the tragic news on his Twitter account late Wednesday night.
"It was really tough warming up, and I thought about him the entire time," he said, holding back tears. "I said that baseball would be a way to clear my mind, but that didn't happen. He was always there."
"I know it is a cliche," he added, "but I really felt like I had someone watching me and helping with that last pitch. That was my best slider of the year."
The A's wore a black patch on their right sleeves that bore the initials "GJN" in honor of Neshek's late son.
"I can't believe the support I've gotten from the team and my teammates," he said. "I couldn't be doing this without that."
Oakland outfielder Josh Reddick said it is "unbelievable" Neshek can still pitch.
"I don't know if I could have even slept if that happened to me," Reddick said.
The A's got off to a fantastic start offensively with Coco Crisp hitting a leadoff homer off Justin Verlander before fizzling at the plate against the reigning AL MVP and Cy Young award winner.
Verlander matched a career playoff-high with 11 strikeouts in 121 pitches against the AL West-championship team that led baseball with 1,387 strikeouts during the regular season.
"It doesn't even seem to matter how many pitches that guy throws," Reddick said. "He just keeps coming after you with all of those pitches."
The upstart A's struck out 14 times overall — striking out three times against relievers Joaquin Benoit and Jose Valverde — and finished with just four hits.
"I think we were a little frustrated, yeah," Melvin said.