Several executions were initially delayed when a judge ruled on Monday that there were still unresolved legal challenges.
The condemned prisoners had argued that lethal injections constitute “cruel and unusual punishments”.
But the Supreme Court voted 5-4 that “executions may proceed as planned”.
Last year, the Trump administration said it would resume federal executions.
Lee was executed by lethal injection in Terre Haute, Indiana, early on Tuesday.
Some relatives of his victims had opposed the execution and sought to have it delayed, arguing that attending it could expose them to coronavirus.
Earlene Peterson, 81, whose daughter, granddaughter and son-in-law were killed by Lee, said she wanted the 47-year-old to be given life in jail, the same sentence as his accomplice.
Who was Daniel Lewis Lee?
Lee was convicted of torturing and killing a family of three in Arkansas in 1996, dumping their bodies in a lake.
Postponed from December, his execution had been rescheduled for 16:00 (20:00 GMT) on 13 July, but was blocked by a ruling from District Judge Tanya Chutkan.
“The court… finds that the likely harm that plaintiffs would suffer if the court does not grant injunctive relief far outweighs any potential harm to defendants,” the judge said.
Her ruling was consequently overruled by the Supreme Court, which said: “The plaintiffs in this case have not made the showing required to justify last-minute intervention by a Federal Court.”