Allen won the first frame but conceded the following three to submit to an early deficit. The 38-year-old did fight back somewhat, but when he found himself 6-3 down in a best-of-19, it looked as though Allen's chances were fading.
However, he rallied to go on to win the next four frames and take the lead. The Northern Irishman then went 8-7 down but held his nerve to claim the next three frames and win the match in impressive fashion.
Earlier in the day, Williams had taken care of Judd Trump in relative comfort. The Welshman had initially raced to a 3-0 lead before Trump surged back to trail just 4-3.
However, Williams won the next three frames to take a stranglehold on the match, and though Trump clawed one back, Williams went on another three-frame run to close out the match 10-4.
On the other side of the draw, Gary Wilson beat Anda Zhang 10-8 in another close quarter-final to set up a semi-final showdown with Ronnie O'Sullivan, who beat Ali Carter 10-2.
