In largely similar fashion to the Magpies' Sky Bet Championship clash at home to Cardiff City on Saturday, Beardsley's youngsters established a healthy lead going into the interval after an encouraging first-half display.
Sean Longstaff and Luke Charman were their marksmen at Twerton Park, the former bouncing back from an early injury scare to net his tenth goal of the season.
A largely low-key second period followed, only for Diego De Girolamo to strike five minutes from time to set up something of a grandstand finale.
Beardsley insisted he remained unconcerned late on, however, believing his charges to have been in complete control throughout.
"(Goalkeeper) Nathan Harker hasn't had a save to make, and in the end it was probably comfortable," he told nufc.co.uk. "Cardiff scored with ten minutes to go on Saturday, but are you really that nervous? I wasn't. I didn't consider it nervy today because I never thought we were going to relinquish the lead.
"We played really well in the first-half and scored two really good goals. We had another couple of chances we maybe should have took, but apart from that, it was mostly positive.
"They're in a good place now. They have done really well over the last month and their mentality is good."
Today's win - their third in succession in all competitions - sees United move to the summit of Group A in the Premier League Cup.
They came up against an experienced City side at Twerton Park - seven of the Robins' starting eleven having figured for the club at first team level.
"Newcastle United are a big scalp for any club - especially Bristol City, with all due respect - but the shoe was on the other foot today because they had a lot of senior players (on show)," Beardsley said. "So, it was a scalp for us to come and win.
"It's brilliant to be at the top of the group. As I've said before, cup competitions are what we want to try and win. Look at the first team, they go to Hull (in the EFL Cup) in a couple of weeks, and hopefully they can keep their run going.
"For us, it's good experience in terms of how to manage a game and how to hold onto a lead, and we did that comfortably today. That was really pleasing."
Beardsley also praised in-form Longstaff for another impressive display, the forward taking his tally for the season to ten with the Magpies' opener at Twerton Park.
The 19-year-old's participation appeared to be in jeopardy just five minutes in after he was left in a heap by the hosts' Kodi Lyons-Foster - a challenge which left Beardsley annoyed.
"I thought it was a horrible tackle," he said. "I haven't seen it again, but their player was lucky that it was the first two or three minutes of the game, so he got away with one for me. You don't want anyone to get sent off, but when it's really nasty - as it was - the referee has to do something about it.
"Sometimes you do get kicked, and the only way to come back from that is to do something good. You can't always score, but Sean's goal was really good. He is in a rich vein of form and he is playing really well."
