Earlier, Alexander Povetkin knocked out previously undefeated challenger Andrzej Wawrzyk in the third round to retain his WBA heavyweight crown, setting up a long-awaited clash with super champion Vladimir Klitschko later this year.
Lebedev, who knocked out previously undefeated Colombian challenger Santander Silgado in his last fight at the same Moscow arena in December, took the initiative from the start.
But 41-year-old Jones was not about to back down. Heavy punishment finally took its toll on the Russian as his right eye was completely shut down by round five.
The men continued their slugfest in the later rounds before the Panamanian stunned Lebedev and the partisan home crowd at Moscow's Crocus City Hall with a right cross as the referee stopped the fight late in the penultimate round.
Jones improved his record to 39 wins, including 31 by knockout, with three defeats and two draws, while Lebedev's record dropped to 25 wins and two loses.
Watched by Klitschko's elder brother and WBC champion Vitaly, Povetkin floored lanky Pole Wawrzyk with a right hook in round two, then sent his opponent to the canvas again in the following round before finishing the job a few seconds later.
The 33-year-old Russian improved his record to 26-0, including 18 by knockout, while Wawrzyk suffered his first defeat in 28 professional fights.
"It was a good workout. I didn't aim for an early knockout, it just happened," Povetkin told Russian television.
Asked if he was now ready to fight Ukrainian Klitschko, who stopped Italian Francesco Pianeta in his last fight in Germany earlier this month, Povetkin said: "Yes, let's do it."
Povetkin's manager Vladimir Khryunov, who last month won the bid to stage the Klitschko fight, added: "It will be held here in Moscow, probably in early October."
Vitaly Klitschko, however, was a bit more cautious.
"The fight is on the cards but the contract has not been signed yet," he told Russian television.
(Reporting by Gennady Fyodorov, editing by Pritha Sarkar)