In a joint statement, the two Palestinian groups responsible for the attack described it as "a natural response to savage aggression by the occupation [by Israel]".
The Israeli defence minister, Amir Peretz, said the attackers were trying to disrupt a fragile two-month-old ceasefire between Israel and Palestinians in Gaza.
A spokesman for Al-Aqsa Martyrs Brigades identified the bomber as Mohammed Faisal Siksik from Gaza City, a member of a faction of the group called the Army of Believers.
At the family home in the town of Beit Lahiya, his mother Rueida said: "He left home on Saturday morning and told me he was going be a martyr.
"I hugged him, cried and prayed that he would succeed in his martyrdom operation. God heard my call."
Mohammed's brother, Naim, agreed. "We can hold our heads high after this brave martyr operation," he said.
Islamic Jihad said the bomber had travelled through Jordan to reach Israel to carry out the attack.
The Jordan authorities denied this and Israel said the attacker had crossed via the Egyptian border.
Rueida, 42, said she was proud that her son "had inflicted harm against the enemy and not died in shameful" inter-Palestinian clashes.
|The bomber who struck the Eilat cafe-bakery|
may have come in through Jordan [Reuters]
Gaza and the West Bank have been rocked by the worst Palestinian infighting since the election of Hamas a year ago and the reactions of the Fatah and Hamas to the bombing underlined the divide between the two.
Fawzi Barhoum, a Hamas spokesman in Gaza, called Monday's attack a "natural response" to Israeli military policies in the West Bank and Gaza Strip.
"So long as there is occupation, resistance is legitimate," he said.
The Al Aqsa Martyrs' Brigades,are linked to Fatah.
A Fatah spokesman, Ahmad Abdul Rahman, condemned the violence, saying, "We are against any operation that targets civilians, Israelis or Palestinians."
The blast took place the Lechamim bakery in a residential neighbourhood of the Red Sea resort town of Eilat, far from its beach hotels, at around 09:40 (07:40 GMT).
The bakery owner and two employees were killed in addition to the bomber, who entered the shop carrying the explosives in a backpack, police said.
Rescuers dressed in white suits spent hours collecting human remains after the blast, which destroyed the inside of the bakery.