While that's a very weak claim (D&D, for most of its 40 year history, suffering from pretty poor class balance at most levels, and generally balanced even worse at higher levels), I still find it difficult to believe.
What's 'lots' and 'high' level in this statement? Were these white room combats, or part of an actual campaign where everyone played up from 1st?
So you meant 'better balanced' than editions that had spells as a less-limited resource than in 5e.
So, 3.5/Pathfinder.
(Classic D&D may have given high-level casters more high-level slots, but they gave lower-level casters even fewer than 5e, and hand no cantrips or short-rest spell recovery, and they piled restrictions on spells and the act of casting, itself, that made them a much more heavily-limited, less flexible resource than they are in 5e.)
"5e better balanced at high level than 3.5" is a much more plausible headline.
That's what theorycrafting has suggested, so far, yes. The reality in play, I've seen though, while only at lower levels, already diverges from it slightly. Caster-martial synergy, such as Hold Person followed up by Sneak Attacks - seems to deliver the most single-target DPR, in actual practice, for instance.
I wouldn't be surprised if having a martial 'blocker' or 3 helps casters output the AE DPR more efficiently, too. Well, or some sort of blocker (companion creatures, summoned monsters, animated dead, illusions, wall spells,etc), anyway.