Norris only narrowly missed the podium that weekend but went on to finish second in the British Grand Prix seven days later.
Once Piastri received the full upgrade, his performances also improved, highlighted by his pole position and victory in the F1 Sprint in Qatar.
It began to paint an interesting picture within McLaren as Norris appeared increasingly frustrated in the final rounds of the year while his team-mate enjoyed a strong run of form.
“I think it's something that we're well aware of,” Brown told Speedcafe on today's episode of the KTM Summer Grill.
“It's something that I think we're particularly good at. That's an area that Andrea and I chat about all the time.
“I very much enjoy working with the drivers, I think if you look at, out of the McLaren environment, [Fernando] Alonso and Carlos [Sainz], and Daniel [Ricciardo], even though unfortunately it didn't work out, we had a great relationship.
“So I think driver management is one of our strengths.
“Of course, they want to beat each other,” he added of Piastri and Norris.
“Right now, they know, pushing each other is going to push the team forward, but for sure there's going to be a day, hopefully sooner rather than later, that they're running one-two, and whoever's in second is going to want to be in first.”
Throughout 2023, McLaren made no secret of the fact the team came first, with decisions taken in race designed to maximise that performance over an individual result.
That saw both drivers moved aside at points or, in other instances, told to hold station where there was little or nothing to be gained by making a change.
There was, however, one clash. As Piastri exited the pits at the Italian Grand Prix, the Australian slid into Norris at the opening chicane.
It was an innocuous clash but prompted a post-race meeting, described by all involved as well-mannered and productive, to understand the hows and whys.
“It does make you nervous, but they're professionals,” Brown said of the prospect of the pair racing wheel-to-wheel.
“They both have a long way to go in their career.
“I'm not anticipating anything silly to happen – and if it does happen, you tackle it right away, you don't let it bubble.
“You see these troubles bubbling up and you kind of question why the teams din't step in early.
“It's kind of like watching two dogs starting to growl at each other on the playground – you know a fight's about ready to happen but yet you stood there and just watched.”
Brown's attitude is motivated by the fact Formula 1 is geared towards rewarding success in the constructors' championship.
Prize money, pit order, and more are all paid out based on the order of last year's points standings.
It's a fact that can create conflict as an individual driver pursues individual glory often in contrast to the team's ambitions.
“From a team perspective, that's what's most important,” Brown said.
“We want to win the constructors' and we want to finish first and second – and we kind of don't care who finishes first and who finishes second.
“Obviously the drivers do, but you race as a team.
“But of course, you know, when we looked at the last race of the year, we weren't in a position not do it, but we did have a discussion internally that, if we can help Lando, without compromising Oscar, to get fourth in the championship instead of sixth, depending on how they were running, we would have done that.
“Oscar was ready to play ball but the race didn't play out in that way.”