The Welshman won the opening frame but Robertson leveled matters with a superb break of 126 before going into a 2-1 lead. Williams then turned the game around to lead 3-2 but Robertson forced a decider in the first to four-frame contest, and what a deciding frame it was.
The World Number One looked certain to advance after carving out a decent lead, but Williams potted a crucial red to set up his revival and a break of 42 saw him take the deciding frame 71-52.
Blackwood's Andrew Pagett will also be hoping to reach the next round when he plays defending champion Stephen Maguire tonight.
Sadly, it wasn’t such good news for Stevens as the World Number 19 lost out 4-3 against Joe Perry. The Welshman won the first frame with a break of 78 and was on the verge of the next round when he took a 3-2 advantage.
However, Perry struck back to level things and then went on to win the decisive frame 71-7 to set up a tie against either Stuart Bingham or Michael Holt.
Elsewhere, Blackwood born Andrew Pagett will be aiming to become the second Welshman to make it into the last 16 when he faces defending Welsh Open Champion Stephen Maguire at the Newport Centre tonight.
Who do you think will win the Welsh Open this year?