I'm with you in that running this stuff in a browser makes me uneasy, although between https and actually trusting Agile Bits, I've launched the 1password web app on a couple of occasions. The vast majority of my use of 1password is their desktop and iOS apps.
The security of their ecosystem is written up in a PDF whitepaper, which is a good read: https://1password.com/teams/white-paper/
tl;dr your master password is never sent to Agile Bits, and they have no way of decrypting the data on their servers.
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