Sounds to me like you inadvertently produced a natural harmonic, albeit for a fraction of a second. It seems that you pluck near to the bridge, which will always produce harmonics best, and the trebly tone helps too.
Harmonics are made when a string is touched gently on a node point. At the bottom end of the string, they're more difficult to produce accurately, so well done!
Try touching the string over the 12th fret, and plucking near to the bridge. You'll hear a ringing sound an octave higher than the open string. There are two other easy ones, above 7th and 5th frets.
As you touch getting closer to the nut, there are more, getting higher each time you move the finger to the left. One of those is your 'culprit'. Because you moved your finger along the string just after, it got killed, but to make it last, take your finger tip away. Or not - as at that point, the string is not moving anyway, so it really doesn't matter. Prove this at 12th fret. Touch, play, then touch again. The harmonic will still ring out!
EDIT: It bugged me a little, so on replay, I guess it might even be a pinched harmonic, produced by simultaneously touching the A string at a point about 2 frets away from the bridge. Sounds weird - but count the first 2 and a bit frets, and measure that distance from the bridge. Touch with a finger/thumb, and pluck with a thumb/finger. Whilst pressing down fret 3! You won't be so lucky now - it'll take a bit of fiddling, especially to find the exact spot for that node. Move along the string slowly.