It came after the US Supreme Court rejected a challenge arguing Arkansas was unfairly rushing several executions before its supply of a lethal injection drug expires at the end of April.
On Thursday the state’s Supreme Court overturned a ruling blocking the use of a different drug.
Lee was convicted for murder and recently told the BBC he was innocent.
His third request to stay the execution was denied.
He was pronounced dead at 23:56 local time on Thursday (04:56 GMT on Friday) at the state’s death chamber in its Cummins Unit prison, a Department of Corrections spokesman said. His death warrant was due to expire at midnight.
Lee did not make a final statement. Instead of a last meal, he asked to receive communion, an official said.
He had been on death row for more than 20 years after he was convicted of beating Debra Reese to death with a tyre iron in 1993.
Her family “had waited 24 years to see justice done”, Arkansas Attorney General Deborah Rutledge said in a statement.
“I pray this lawful execution helps bring closure for the Reese family,” she said.
Lee told the BBC’s Aleem Maqbool in a recent interview that he was innocent, and death row was like a “living nightmare”.
The state had initially planned to carry out eight executions in 11 days, before the 30 April expiration date of its supply of the drug, midazolam – used together in lethal injections with two other drugs.
The first three executions were cancelled due to various court rulings.
The other inmate due to die on Thursday has been given a stay to make time for advanced DNA testing that his lawyers say could prove his innocence.
Stacey Johnson was convicted of the murder of Carol Heath, who was beaten and had her throat slit in her flat in 1993.
Thursday saw a flurry of legal activity, with the new Supreme Court Justice Neil Gorsuch siding with conservatives in the 5-4 decision to reject the inmates’ claim that the accelerated process amounted to “cruel and unusual punishment”.
It followed Arkansas’s Supreme Court’s overturning of a ruling by a lower court that blocked the use of the drug vecuronium bromide.
The McKesson Corporation, which supplied the drug, had accused the Arkansas Department of Correction of failing to say it planned to use the drug for executions.
Like many US states, Arkansas has struggled to find the drugs it needs to carry out executions. Its last was in 2005.