Explaining academic probation as an undergrad on PhD application, after doing well in masters
I think you know the answer to this: it's what you just told us. If I read that on an application, I'd nod and keep going, my concerns fully assuaged. (I've read similar, and still recommended admission of the applicants in question.)

If, however, I saw a bunch of lousy grades and a probation notation on an undergraduate transcript with no explanation of what happened and what's different now, that would be a huge red flag for me even considering the better graduate-school record. I would wonder whether the applicant would crash and burn again, and what else the applicant might be hiding from us.

Face it head-on, the way you just did here. I can't guarantee it'll be fine, but I think it's by far the best of your available alternatives. People aren't perfect and they grow over time; we know that, and a lot of us are not averse to second chances. If you learned from what went wrong and put it behind you, that's what we need to know.