score:5
Yes, you can do this, and no, there is no need to tell anyone that you have done it.
A US Visa is only relevant as an entry document - once you are in the US it is no longer relevant, and it is your I94 (either physical, or as they generally are now, the relevant entry on a computer) that dictates your status in the US.
When you renew your passport, both your US Visa (in the old passport), and your I94 remain current up until their relevant expiry dates.
There is one potential issue, which is the one you've mentioned - your I94 exit record (on the new passport) may not be correctly tied to your entry record (on the old passport). In theory this would never become a problem as they will automatically link the two passports based on your personal details, and most importantly based on your fingerprints - however if for some reason this didn't happen automatically it would be a trivial exercise to point out what had occurred and have them match the relevant entry and exit records together.
Personally I've renewed my Australian passport twice whilst in the US on a visa, and I've never had any issues doing so, or when subsequently leaving/entering the US on future visits. The only difference I noticed was on the first entry using the new passport they took fingerprints from all 10 fingers, instead of the 4 they normally take when I enter the country. According to the immigration official who did this last time I entered with a new passport, this was so they could confirm the linkage between the old and new passports.