At least five people were killed and more than 30 were hurt in the blast near a government building on Saturday.
The nationalities of the dead are Israeli, American and Iranian.
Brendan Ward, Irish Ambassador to Turkey said two adults from the Irish-Algerian family were slightly hurt, but their three children escaped injury.
Mr Ward told the Irish state broadcaster RTÃ‰ that the family was on the final day of a holiday in Turkey when they were caught up in the explosion.
He said the two adults were treated in hospital but have since been discharged.
"I understand that they are leaving [Turkey] this evening on their flight back to Britain where they live. Their injuries were relatively minor," Mr Ward added.
"Thankfully, one of the embassy's staff was in Istanbul and was able to facilitate them through translation and completing formalities with the police and the hospital."
Mr Ward said terrorism "remains a problem" in large cities in Turkey and said advice was available on the Irish Department of Foreign Affairs website for anyone planning to travel to the country.
Earlier, the Irish Minister for Foreign Affairs Charlie Flanagan said he was "deeply saddened" by the "horrific" attack.
In a statement, he said: "I have spoken to Ambassador Brendan Ward in Turkey and can confirm that we are aware of a number of Irish citizens among the injured.
"An embassy official is on the ground in Istanbul to provide consular assistance.
"Anyone with concerns for Irish citizens please ring 01 4082000 or Embassy Ankara on 0090 312 4591000. If you are in the area, please follow the instructions of the police and local authorities."
On Sunday, Turkey's interior minister said the suicide bomber belonged to so-called Islamic State (IS).
Saturday's attack in Istanbul - Turkey's largest city - happened at about 11:00 local time (09:00 GMT).
An Irish-Algerian family was among those injured in a suicide bomb attack in Istanbul, according to the Irish ambassador to Turkey.