Simpson was too strong for Botswana’s Alister Walker and raced through in 42 minutes, winning 11-2, 13-11, 11-9.
Willstrop had more to worry about against fellow Englishman Jamie Haycocks as he dropped two games in a match lasting over an hour.
Matthew, seeded fourth, also faced difficulty against stubborn New Zealander Campbell Grayson.
The Sheffield player dropped the second game and was then made to fight for victory in a mammoth fourth as he won 11-7, 6-11, 11-4, 13-11 in 57 minutes.
Matthew said: “I started maybe too well and he was slow and I was thinking it was easy and then all of a sudden I was in a battle.
“I had to dig it out there in the fourth and really happy it didn’t go to five.”
Willstrop, who reached the semi-finals 12 months ago, took the opening game 11/9 but Haycocks battled back to take the next by the same score.
A comfortable third went Willstrop’s way but Haycock’s found a response to take it into a decider.
But Yorkshireman Willstrop found another gear in the fifth to take it 11/4 and wrap up a 11/9, 9/11, 11/5, 9/11, 11/4 win.
Willstrop said: “Jaymie played well, he had no worries in the world, no pressure, he just played his squash and made it very hard for me. “There’s not much to say really, very happy to be able to compete still, truly happy to win.”
The world No 19 will now face Egypt’s Ali Farag in round two after the tenth seed battled past England’s Declan James.
Simpson, who lives in Harrogate, will play 16th seed Max Lee of Hong Kong on Friday.
Simpson said: It’s the biggest tournament in the world, I made my big breakthrough in 2012 and since then it’s been a really disappointing tournament for me.
“I’m really happy to get through to the second round, Alister’s a training partner of mine from when I first turned pro and I’ve never beaten him, so I’m really, really happy.
“I think the biggest thing in the second and third games was Alister’s level. I felt like he wasn’t up to pace in the first game, but knew he would improve in the second and play better.
“I was very lucky to win that game, I saved four match balls.
“Looking around at some of the results today, these results are very difficult to play on and it’s not easy to beat anyone 3-0 these days, so it’s great to conserve that energy.”
Sheffield’s Nick Matthew, seeded fourth, faces Campbell Grayson later on Thursday.