Fans at the famous venue, which seats more than 45,000, were left stunned when Nery floored the unbeaten "Monster" in the first round with a huge left hand.
But Inoue got up and knocked his Mexican opponent down in the following round and sent him to the canvas again in the fifth before finishing him off with a right hook in the sixth.
"It was a great feeling to knock him down but how about that surprise in the first round?" said the 31-year-old Inoue.
"As a boxer, when that happens to you it fires you up. It gave me a lot of energy."
It was the first boxing event to be held at the Tokyo Dome since 42-1 underdog James "Buster" Douglas knocked out unbeaten heavyweight champion Mike Tyson in February 1990 in one of the sport's biggest upsets.
"There was a lot of pressure on me as the main event in the first boxing at Tokyo Dome in 34 years, but it gave me strength," said Inoue.
Inoue took his record to 27-0 with 24 wins by knockout.
It was his first fight since becoming the undisputed super-bantamweight world champion in December.
He is just the second man to become undisputed world champion at two different weights since the four-belt era began in 2004. American Terence Crawford was the first.
Rapturous welcome
Fans at the Tokyo Dome gave the 31-year-old a rapturous welcome as he entered the cavernous venue with all four world championship belts up for grabs.
A huge upset looked like it could be on the cards when Nery sent the champion sprawling with a massive left hand.
But it seemed to energise Inoue and he came flying out of his corner to send Nery down in the second round.
Inoue continued to land ferocious blows and he sent his opponent to the canvas again with a left hook in the fifth round.
The referee stopped the fight after Inoue downed Nery for a third time in the sixth round.
Nery, a former two-division world champion, was fighting in Japan for the first time since he was stripped of his bantamweight world title in 2018 after failing to make weight for a bout there.
He was subsequently banned from fighting in Japan but the country's boxing commission gave him the green light to compete against Inoue.
Nery was booed when he was introduced to the Tokyo Dome crowd.
The 29-year-old's record fell to 35-2, with 27 KOs.
'A real struggle'
Japan's Yoshiki Takei beat Australian defending champion Jason Moloney by unanimous decision to win the WBO bantamweight world title on the undercard.
Takei, a former kickboxer who had won all eight of his previous boxing matches by knockout, started strongly before Moloney came back into the bout in his second title defence.
"The 12th round was a real struggle," said the 27-year-old Takei.
"I wanted to win by knock-out but Moloney was really tough."
Inoue's younger brother Takuma defended his WBA bantamweight title with a unanimous decision win over fellow Japanese Sho Ishida earlier on the undercard.
Ishida floored Takuma Inoue in the first round with a left jab but the champion recovered to dominate the fight.
Japan's Seigo Yuri Akui defended his WBA flyweight title with a unanimous decision win over countryman Taku Kuwahara.
