An Israeli army spokeswoman said there were four Israeli casualties in the attack at the Kerem Shalom border crossing on Sunday morning.
After the attack, a small Israeli force mounted a search operation in the southern Gaza Strip near the scene of the incident to look for the missing soldier.
Israel's military chief of staff said he believed the missing soldier was alive.
Officials from the Popular Resistance Committees (PRC), a small Palestinian group, said it carried out the assault as revenge for the Israeli assassination of the group's founder, Jamal Abu Samhadana, earlier this month. The PRC said that members of Hamas's military wing had also participated.
The incident, in which PRC fighters managed to get into Israel, came days after a warning was given of an imminent attack in the area. The dawn raid was the first carried out inside Israel from the Gaza Strip since Israel withdrew from the occupied territory in September last year.
The army spokeswoman said: "Apparently an anti-tank missile hit a tank. The clash is still in progress."
The Israeli army said there was an exchange of fire near the crossing, which is used to bring cargo from Egypt into Gaza.
Aljazeera received a statement saying that a number of Palestinian resistance fighters had been dropped behind the separation fence.
However, the Israeli army said the fighters tunnelled under the border, and fired rocket-propelled grenades and automatic weapons.
Israeli soldiers patrol the area
near the border crossing
An Israeli military source said seven gunmen infiltrated and attacked an armoured personnel carrier.
Speaking to Aljazeera, Khalid al-Batsh, an Islamic Jihad leader, congratulated the Palestinians for the "brave operation".
"The ongoing Israeli aggression and crimes against Palestinians confirm that our people would never give up and would continue resisting the occupation," he said.
Abu Mujahid, a PRC spokesman, said: "Our fighters infiltrated the Israeli army military location near so-called Kerem Shalom.
"They succeeded in blowing up several Israeli vehicles and clashed with Israeli soldiers. The battle is still going on. The number of fighters is bigger than any time. We have some martyrs who fell during the battle."
He said full details of the operation would be released at a news conference later in the day.
The operation, he said, was meant to avenge the death of the group's leader, Abu Samhadana, who was killed by an Israeli air attack on June 8. Abu Samhadana was killed shortly after accepting a senior security position in the Hamas-led government.
The Gaza-Egypt border was closed after the attack.
Palestinian officials said the European observers who oversee the Rafah international border told them they would not be opening the border on Sunday because they could not get into Gaza from Kerem Shalom, the crossing used by the monitors to get from Israel to Gaza.
Israeli soldiers in a security drill
at the border (File photo)
Israel had closed the Rafah border, Gaza's only gateway to the world, for much of the past four days citing security concerns.
The Israeli army issued warnings over several days last week that militants were plotting an attack at Kerem Shalom. The warning prompted EU monitors to close the nearby Rafah crossing, Gaza's main gateway to the outside world.
Hamas ended its 16-month-old truce on June 9 after seven members of one Palestinian family were killed on a Gaza beach during a day of heavy Israeli shelling. Hamas has blamed Israel for those deaths. Israel has denied responsibility.
Israel completed a withdrawal of Jewish settlers and soldiers from Gaza in September last year after 38 years of occupation. No Israeli soldiers have been killed in Gaza since then.
Mahmoud Abbas, the Palestinian president, condemned the attack in a statement saying: "We have always warned against the danger of certain groups or factions leaving the national concensus and carrying out operations for which the Palestinian people will always have to pay the price."
Israel closed all crossing points leading to Gaza.