The 1997 champion Ken Doherty then held off a determined comeback from John Higgins, who was 10-0 down at one stage, to book his place in the last four with a 13-8 victory.
A delighted Williams said: “To beat Stephen 13-7 is a good result for me. He can play a whole lot better than that.
“I played okay throughout the match really and did enough to win in the end.”
The victory also saw Williams reclaim the world No.1 spot from Ronnie O’Sullivan, repeating the feat only previously achieved by fellow Welshman Ray Reardon.
“It’s a great achievement for me and it means a lot,” said Williams, who feels he is playing at the same level as when he won the world title three years ago.
“I’m playing just as good and have been for the last six to 12 months,” he added. “But I know I can play a lot better again.”
Hendry described the result as a “massive disappointment” as his bid to land an eighth world title ended in failure.
He said: “I just didn’t take enough of my chances to win and lost to the better player on the day, it’s as simple as that.”
Williams will face the winner of the match between Stephen Lee and Marco Fu in the semifinals. Lee was leading Fu 8-4.
Higgins clawed his way back by winning seven frames in a row to trail Doherty by just three frames at 10-7.
But his hopes of completing a remarkable recovery ended when Doherty won three out of the next four frames to seal victory.
A relieved Doherty said: “I could just see the headlines, ‘The Biggest Collapse Since Wall Street’. The match turned completely but I just held it together.”
Doherty could not believe he was leading 8-0 at the end of the first session. “I think I played the best eight frames I’ve ever played,” he said.
Doherty will play either defending champion Peter Ebdon or Paul Hunter in the last four. The pair were level at 8-8.
Quarterfinal results and scores:
Mark Williams (Wales) bt Stephen Hendry (Scotland) 13-7. Frame scores (Williams first): 55-57 96-23 131-1 64-65 115-5 0-76 89-1 77-64 28-75 64-34 5-71 72-39 1-111 82-12 70-33 80-0 46-92 81-0 62-22 71-44.
Ken Doherty (Ireland) bt John Higgins (Scotland) 13-8. Frame scores (Higgins first): 9-62 16-85 0-107 65-66 1-66 44-83 58-71 7-100 9-64 9-112 117-0 70-24 74-12 64-52 67-57 62-57 69-31 56-68 7-75 94-8 1-71.
Stephen Lee (England) leads Marco Fu (Hong Kong) 8-4. Frame scores (Lee first): 45-65 0-132 74-57 46-60 76-15 122-20 83-25 4-119 86-24 88-8 67-43 51-7 - interval
Peter Ebdon (England) and Paul Hunter (England) level at 8-8. Frame scores (Ebdon first): 0-89 63-70 95-13 98-0 (98) 13-61 78-0 4-87 60-58 73-8 100-30 8-120 62-16 13-71 73-28 0-129
Semifinals: May 1-3 (best of 31 frames)
Final: May 4-5 (best of 35 frames)—Reuters