The three prisoners, all members of the Islamic State group, were serving life sentences at the Taji penitentiary north of Baghdad, the security services said in a statement.
They were spotted as they tried to break out of jail by climbing over an external wall, the statement said.
Guards opened fire "when they refused to heed warnings", it said, adding one prisoner was killed while the two others "surrendered".
"The three terrorists had been sentenced to life in jail," the statement said without identifying them.
The Islamic State group swept across swathes of Iraq and neighbouring Syria in 2014 where they set up so-called caliphate.
Iraq officially declared victory over IS in 2017, and two years later they were defeated in Syria.
But sleeper cells continue to be active in both countries where they frequently carry out attacks.
After the demise of IS in Iraq, courts in the country have sentenced hundreds to death for crimes perpetrated by the jihadists.
Only a small proportion of the sentences have been carried out, as they must be approved by the president.
Barham Saleh, who has held the post since 2018, is known to be against capital punishment.