Boccieri quickly left the podium while the former president addressed a crowd of about 1,000 people in Canton, about an hour south of Cleveland.
"The baby is now being born!" Clinton announced as the crowd erupted with cheers. "You'd be amazed how many times I take a picture with a very pregnant woman and then she immediately gives birth."
Boccieri, a Democrat, is locked in a closely watched race with Republican Jim Renacci.
Addressing both sets of grandparents, who were among the crowd standing in the parking lot, Clinton joked: "I'd like some credit for your fifth grandchild being brought into the world."
Clinton couldn't resist one more wisecrack before continuing his speech.
"We got another Democrat," he said. "I wish we could register that baby before it's too late."