Newcastle ended a 70-year domestic trophy drought by beating Liverpool to win last season's competition and once Joelinton and Osula scored in the space of two minutes early they were never in any danger of a shock exit.
The night could have been billed Premier League v League One as Manchester City, Arsenal and Tottenham Hotspur all entered the competition against third-tier opposition.
Phil Foden was on target for Manchester City as they beat League One Huddersfield Town 2-0 away while Eberechi Eze netted his first goal for Arsenal since his move from Crystal Palace to send his side to a 2-0 win at Port Vale.
Tottenham Hotspur's bright start to the season under new manager Thomas Frank continued as they cruised to a 3-0 victory at home to Doncaster Rovers.
The fourth-round draw, made late in the evening, gave Newcastle a home tie against Tottenham while Manchester City are away at second-tier Swansea City. Arsenal will host Brighton and Hove Albion while Liverpool are at home to Crystal Palace. Chelsea are away at Wolverhampton Wanderers.
Newcastle manager Eddie Howe fielded a strong line-up for his side's opening hurdle in the defence of their trophy and his team proved too slick for League One leaders Bradford.
Joelinton put them ahead in the 17th minute with his first goal for nine months, controlling Anthony Gordon's shot and placing his effort past Bradford keeper Sam Walker.
Osula ran through to make it 2-0 in the 19th minute but Newcastle had to wait until the 75th minute for Joelinton to make it 3-0 with another cool slotted finish.
Boyhood Newcastle fan Andy Cook smashed a reply for the visitors after coming off the bench but Osula confirmed Newcastle's superiority with his second goal.
Foden has shown glimpses of a return to his best form for City and he gave them an early lead with a clinical finish after exchanging passes with Divine Mukasa. He then set up Savinho in the 74th minute to wrap up the tie.
Eze made a big impact when coming off the bench against Manchester City in the Premier League on Sunday -- his pass creating a stoppage-time equaliser for Gabriel Martinelli.
On Wednesday, Martinelli returned the favour, helping set him up to score in the eighth minute.
"Very special. I've been waiting to score for this club for a long time and the first is very special," Eze said.
Arsenal were rarely in top gear and needed a late goal by Leandro Trossard to wrap it up.
Joao Palhinha scored with an acrobatic overhead kick to give Tottenham an early breakthrough at home to Doncaster, his first goal for the club and the hosts were in cruise control after a Jay McGrath own goal. Brennan Johnson got on the scoresheet for the London club in stoppage time.
The draw for the last-16 included three Welsh clubs for the first time and two of them meet, with Hollywood-backed Wrexham at home against Cardiff City.
Forest denied win
Nottingham Forest’s long-awaited return to European competition ended in a 2-2 Europa League draw at Real Betis, with Igor Jesus’s first-half double cancelled out by a late Antony strike, while AS Roma opened their campaign with a 2-1 victory at Nice.
Forest’s return to a major European competition for the first time since March 1996 began on the wrong foot, with Betis taking the lead inside 15 minutes when Cedric Bakambu fired an effort high into the net.
The two-time European Cup winners responded swiftly as Igor Jesus levelled from close range within two minutes and then headed in again in the 23rd minute.
Both sides pushed for a breakthrough following the interval. Betis gradually seized more control as the match wore on and were finally rewarded five minutes from time, when Antony tucked away Marc Roca’s cross to ensure the spoils were shared.
With the result, Forest boss Ange Postecoglou became the club’s first permanent manager since Billy Walker in 1946 to go without a win in his opening four matches.
"I think first I should be proud of the players' efforts. It's a difficult place to come, difficult conditions, people don't realise the humidity out there," Postecoglou said to TNT Sports.
"I'm just disappointed that the players and supporters don't get the rewards for our efforts. I've just got to make sure they keep their heads up because there's plenty to be positive about and the wins will come."
New Zealand striker Chris Wood did not feature in the match.
