I'm using TKlerx/android-sqlite-browser-for-eclipse for instant browsing inside Android Device Monitor. I found it the fastest and most practical way.
Just download the jar from the releases page and place it in the [android-sdk-folder]/tools/lib/monitor-x86_64/plugins.
Afterwards, just browse any database file inside the File Explorer and click the blue-ish icon shown in the picture:
Enjoy !
