A masked spokesman for the group, Abu Ubaida, appeared on a Hamas television station to make the announcement.
He said: "We have captured a Zionist soldier and the occupation has not admitted that."
Unconfirmed reports named the captured as Sha'ul Aron.
The Israeli Defence Force is reported to have denied the claim.
The news was met with celebrations in Gaza.
The claim came on a day that has seen over 100 Palestinians killed during Israel's ground offensive into Gaza.
Heavy shelling of the Shejaiya district in the north east of Gaza has been described as a "massacre" by Palestinian President Mahmoud Abbas.
The intensity of the bombardment meant the emergency services could not access the area and dead bodies lay in the streets as thousands fled, the Agence France Presse news agency reported.
The UN Security Council is to hold an emergency meeting at 2:30 AM BST.
A humanitarian truce was called in the area after the attack but violated after just one hour, with both sides blaming each other for the attack, the BBC reported.
Thirteen Israeli soldiers have also been killed. Seven died when their personnel carrier was hit by an anti-tank missile.
Many Gazans have fled their homes but as the conflict continues to escalate, but where they can find shelter is unclear.
Despite the bloodshed and scenes of carnage coming from Gaza, Israel's Prime Minister Bejamin Netanyahu has vowed to continue with the offensive, blaming Hamas for any civilian deaths.
He said: "We regret any civilian deaths but those lay entirely at Hamas’ door.
"What Hamas is doing very cynically is embedding its rocketeers, its rocket cashes, its tunnels — these terror tunnels in homes, in hospitals, in schools, and when we take action, as targeted as we can, they then use their civilians as human shields.
"So Hamas is both targeting civilians and Hamas is hiding behind civilians. That’s a double war crime, and therefore all civilian deaths as regrettable as they are fall on their shoulders."