Flyweight Barnes won his bout against Tanzanian Hamadi Furahisha with a technical knockout in the third round on Monday evening.
Barnes will take on Papua New Guinea's Charles Keama in a quarter-final round on Wednesday.
While welterweight Steven Donnelly defeated Tonga's Oscar Finau in a unanimous decision.
Women's flyweight Michaela Walsh defeated Mauritius's Thessa Dumas to secure her place in the quarter-finals.
Earlier Belfast's Conlan, 22, defeated Shiva Thapa of India on a unanimous decision to seal his last eight slot in the bantamweights.
For the world-ranked number two, it was his second fight of the competition.
"That performance wasn't my best. I was a bit nervous going into that fight. But there is a lot more to come from me," Conlan said.
"He (Thapa) tried to come in and be a bit of a roughhouse which doesn't really work for him. It works more for me."
He added: "Our boxing team is inexperienced, but we are all in reach of medals. The training and work we have put in has been top notch.
"I believe we can do the business, I think there are medals on our team.
"Anything else but gold would be a major disappointment for me. Gold's the only thing I want. Gold's the thing I'm going to get."
Conlan sustained a cut to the head in his previous fight, but definitely hasn't let it stop him.
It's only a wee cut on the head. I've probably had bigger cuts shaving and I don't even shave!
Meanwhile, Derry man Coyle won out over Guyana's Dennis Thomas, getting the nod on the three judges' scorecards.
It all comes after weekend progression for four other Team NI boxers.
Belfast flyweight Ruairi Dalton defeated Tanzania's Ezra Paul Mwanjwango by unanimous decision and will fight Australia's Andrew Moloney on Tuesday.
Speaking about his second successive unanimous decision of the Games, Dalton said: "That's something that gives me a lot of confidence. It's another step closer to a gold medal and I'm certainly in it to win it."
"You don't get much of a break, you are fighting every two days but that's boxing for you, nothing is easy. I'm happy with my performance, I tried a wee bit at the end but I stuck to the plan.
"Everyone is trying to keep the momentum going and for such a small country we are doing really well."
It's brilliant to be in the final eight of the Commonwealth Games and I can't wait to get going again.
There was joy for Joe Fitzpatrick as he beat Lesotho's Qhobosheane Mohlerepe in lightweight last 16 by split decision.
He'll fight Kenya's Okongo Nicholas Okoth in the quarter-finals on Tuesday.
He said: "I was a bit rusty. I got a bye, so I was hanging around for a few days just itching to get in the ring. You get into it quickly, though.
"It was a good fight, but he was a very dirty fighter. I was happy with my performance and the win and the atmosphere was brilliant but I was able to stay in the zone."
The 19-year-old, who is the youngest of Team NI's boxing team, said: "I might only be 19, but George Best was younger than that when he played in front of big crowds so it's no big deal. It can only get better for me."
Sean Duffy took on Tanzania's Gaudence Fabiani Pius, winning his fight by split decision. He faces Malaysia's Akyazlan Khir Azmi on Tuesday.
"Yesterday was the hard one. They call it ring rust. The first fight you have to get used to the surroundings and used to the crowd, it takes a lot out of you. We had to focus on the job at hand today and luckily it is onwards and upwards," said the boxer.
"Our coaching team has got everything covered so there's no danger of us not being fully prepared when we get in the ring. We proved that."
After a slow start, boxer Sean McGlinchy won his heavyweight contest against Jamaica's Cheavan Clarke by split decision.
He will fight Kenya's Ajowi Elly Ochola in the quarter finals.
"After round one, the coaches had to wake me up a bit and say get your boxing shoes on but in the second and third I came out and showed heart and true sportsmanship.
"I feel happy with what I've done tonight apart from the first round and I'm going to bed a happy man tonight," said McGlinchy.
There was disappointment, however, for Delhi silver winner Steven Ward after he lost out on his first fight in the heavyweight division against England's Warren Baister.
I'm just gutted. It was a tough fight and I felt it could have gone either way but I've no one to blame but myself.
"It wasn't the coaches' fault and it wasn't to do with my training, it was the last two rounds," Ward said.
Looking back at his performance, he added: "I would have changed a lot, I would have been busier and a couple of times I was trying to fight when I didn't really have to.
"I felt I won the first round clean by a mile, but I didn't stick to the plan for the next two rounds and I'll learn to listen to my coaches more.
"I wasn't here just to make up the numbers, I wanted to win a medal.
"I put in so much hard work and it's just gutting."
Ward says he is "disappointed" he drew Baister, the highest ranked boxer in the tournament, in the first round.
He added: "This is the first year there's been no seeding and he was the number one in the rankings and I was number two so this should have been a fight for the final, not the first round. It's the fight everyone wanted to see in the final."