Joshua beat China's Zhang Zhilei 15-11 in his late super heavyweight bout after Ogogo had defeated Germany's Stefan Hartel 15-10 in the middleweight division.
The pair are now guaranteed to take at least bronze.
Roared on by a passionate crowd at the ExCel arena in south east London, Ogogo was awarded the first round 5-2 despite the exchanges between the cagey pair looking even. The Polish judge gave the round 7-1 to the Briton.
Undeterred, Hartel continued to press and continually pushed Ogogo back with his strong jab but was restricted after sustaining an injury and the Briton came up with some fast combinations to take the victory.
"In the first round I damaged my right hand, it had already been broken before so that limited me," Hartel, who came into the media area shaking his wrist in pain, said.
"I feel that because of the crowd supporting Ogogo, the judges may have favoured Ogogo."
The Briton accepted he had taken some punishment from the German. He next faces Brazilian Esquiva Falcao Florentino.
"I took a few rights hands so I might get a slap on the wrists tomorrow," Ogogo said before looking ahead to his bout with Florentino.
"I'm boxing the Brazilian that beat me in the world championships but I've got two good arms now so I'm looking forward to doing battle with him," citing an injury as the reason he lost their previous encounter.
Florentino guaranteed Brazil their first men's boxing medal since 1968 with a dominant performance over Hungarian Zoltan Harcsa, which suggested he could win more than just a bronze.
Florentino's 14-10 middleweight victory meant a semi-final place and only a second medal in men's boxing for the football-obsessed nation, who will host the Games in four years time.
"I'm very, very proud to be the first to guarantee a medal for Brazil here," Florentino said. "I'm very proud because I told my dad (I) was going to take a medal home and now (I have)."
"I'm not here to win the bronze medal, I'm here to win the gold medal and I am confident I will get the gold."
The middleweight's medal came a matter of hours after his compatriot Adriana Araujo secured at least a bronze in the women's lightweight category by reaching the last four.
But Brazilian fans will be confident of more than that after Florentino easily produced the most eye-catching display of the four middleweights to advance to the semi-finals.
His lightning quick hand-speed had Harcsa worried and after the Hungarian felt the full force of some thunderous uppercuts he chose to stand back, wary of Florentino's power.
The Brazilian led 6-2 after the first round and fought within himself to edge over the line as he slowed down the pace of his attack and preferred to throw more considered shots.
In the other semi-final Uzbek Abbos Atoev will face Japanese second seed Ryota Murata on Friday after they both scored 17-13 wins in their quarter-finals.
Murata produced a big 10-5 third round to come from behind and take his contest, while Atoev beat Vijender Singh, which left the Indian's corner complaining at what they felt was another wrong decision.
The Indian team are still reeling from the International Amateur Boxing Association (AIBA) decision to change the result of welterweight Vikas Krishan's victory to a defeat by American Errol Spence following a video review and were unhappy again.
"It's a mafia, mafia, mafia, no point complaining," India's Cuban boxing coach Blas Fernandez shouted as he walked to the dressing room.
There was more controversy in the super heavyweight division after Olympic champion Roberto Cammarelle beat Mohammed Arjaoui by the tightest of margins.
The Moroccan's coach wanted to know one thing: was one of the judges Italian?
Defeated fighters and coaches have groaned all week about the scoring in tight contests and the International Boxing Association (AIBA) has already sent two officials home and suspended another for wrongdoing.
Cammarelle, fighting well below par, gave Arjaoui a chance of winning his country's first medal of the Games and while he probably did just edge it as the judges awarded him a 12-11 win, Abdelhak Achik disagreed.
"The judges always get it wrong and we never win. As soon as the fight starts, the judges are against us and they are always against the Arabs," the coach, who flung his fighter's towel to the ground in disgust when the result was announced, said.
"Only two Africans have qualified for the quarter-finals. It's a bad advertisement. Is one of the judges Italian?"
Asked about the Moroccan's charges of a bias against Arab fighters, a spokesman for AIBA told Reuters that such comment did not merit a response.
AIBA also rejected a protest from the Moroccans before even sending it for review.
There was no doubting the result of the fight two bouts later when to the delight of a home crowd who waited patiently until close to midnight, Joshua guaranteed Britain another medal by reaching the semi-finals.
The big Londoner did it in style too, knocking China's Zhang Zhilei flat on his back in the second round with a whopping right hand that brought the packed house to their feet.
"That medal represents a journey, a lot of hard work, but it hasn't stopped here," the world championship silver medallist said after signing autographs on his way out of the arena.
"It is going to get tougher and I'm going to have to keep my heads on my shoulders and try to change the colour of that medal. I am going to walk away from these Olympics a new man."
It gets tougher in the form of third seeded Kazak Ivan Dychko who eased past Simon Kean of Canada 20-6.
Earlier Azerbaijani world number one Magomedrasul Medzhidov bashed his way out of a hole with a big third round against Russian Magomed Omarov in a contest light on tricky footwork but packing plenty of power.
"I knew it was going to be tough but we have a saying in boxing - death or glory," Medzhidov, who next faces Italy's Cammarelle, said.