For starters, that vayne was plat 4 and you didn't have much max hp as you went Ravenous Hydra, netting you just under 3000.
Im going to guess you had around 300 armor with base and runes, meaning you take 75% reduced damage.
vayne had ~200 AD with IE and 100% crit, making every auto do around 450 vs 0 armor, so with 75% reduction brings that to 112.5.
PLUS Statikk shiv and RFC, meaning when it procs it deals (160*2.5)[its modified by crit chance and crit damage] = 400. You likely had around 54 MR, reducing this damage to about 240.
This brings her first auto damage to 350.
Her first 3 autos together do: (113+240)+113+113+(12%*~3000), doing roughly 940 damage. it would take roughly 9 autos to kill you. With vaynes approximate 2.2 attack speed, it would take 4 seconds to kill you.
The math does add up :)