The Scimitar's gunner and two other passengers, who along with the slain soldiers are all from the Queen's Royal Lancers Battle Group, were injured in the attack and are currently being treated for their wounds at Tallil airbase in Dhi Qar province.
Cpl Leaning was commanding the Scimitar and Trooper Turton driving it when the fatal attack occurred at 11:20 local time (08:20 BST) yesterday.
Their deaths mean that 144 UK military personnel have now died in Iraq since the 2003 US-led invasion.
The army today described Scunthorpe-born Cpl Leaning, known to his friends as Bill, as a "fine soldier and natural leader".
A family statement said that the soldier, who was one of the first to be posted to Iraq four years ago, was a "loving son and a great friend".
"We are most proud of the fact that Ben was prepared to do his duty with the utmost professionalism," the statement continued.
"He will be very sadly missed by all who knew him. We now ask to be left in peace to mourn our loss."
Cpl Leaning's commanding officer Lieutenant Colonel Richard Nixon-Eckersall said of the 24-year-old: "He had already made a name for himself at the regiment as one to watch for the future.
"He was the first to volunteer for any course or task, and I have no doubt he would have volunteered to lead the patrol that so tragically saw him killed. His death is a huge loss to the regiment."
Trooper Turton, from Grimsby, joined the regiment in 2003, and had been marked out for specialist duties due to his martial skill.
A trained sharp shooter and assault pioneer specialising in demolitions, the army describes him as an "exceptional soldier".
His wife Sharon said in a statement: "Kris always told me he lived for two things – me and the army. I am proud he has died doing something he loved so much. He was the most amazing person I have ever met and I loved him with all my heart and soul as he loved me.
"He was kind, generous and was always able to make everyone smile with his brilliant sense of humour. He always walked with dignity and he will never walk alone."
Lt Col Nixon Eckersall added that Trooper Turton was "selfless and committed" and "strived for perfection in everything he did".
"He quickly made an impression as a future NCO and I was particularly pleased with how he had embraced the dismounted aspect of our job. Our thoughts are with his wife Sharon and all of his family. His loss will be keenly felt across the battle group," he went on to say.
Commenting on the deaths, defence secretary Des Browne said: "It was with profound sadness that I learned of the deaths of Cpl Leaning and Trooper Turton on operations in Iraq yesterday.
"These young men died serving our country and helping to bring peace to Iraq. They will be deeply missed. Their families and comrades in the Queen's Royal Lancers are in my thoughts."
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