Clayton registered a hat-trick of ton-plus finishes against English debutant Lipscombe to book his place in the last 32.
"It's a very nervy first round, and Adam is a great player," he said. "If I can finish like I did tonight on the last set, I'll be okay.
"If I can play my game, they've got a tough cookie to crack," added Clayton, discussing his progression to the second round, where he'll face Dom Taylor. "Hopefully it's going to be my year."
In other first-round clashes, Martin Lukeman lost 3-1 to Max Hopp, and 29th seed Dirk van Duijvenbode beat Andy Baetens 3-2.
Connor Scutt, meanwhile, fended off a spirited Simon Whitlock comeback. The Englishman led 2-0, but was pushed to five sets by the Australian for a memorable 3-2 win.
Evening results
Martin Lukeman 1-3 Max Hopp
Dirk van Duijvenbode 3-2 Andy Baetens
Jonny Clayton 3-1 Adam Lipscombe
Connor Scutt 3-2 Simon Whitlock
Wright cruises as Menzies loses temper
Two-time champion Peter Wright also progressed earlier in the day with a 3-0 victory over Noa-Lynn van Leuven, relying on accurate finishing despite a modest scoring average of 83.51.
Wright seized an early break of throw and punished Van Leuven’s erratic doubling to secure a straight-sets triumph.
In a contrasting first-round encounter, fellow Scot Cameron Menzies suffered a dramatic 3-2 defeat at the hands of debutant Charlie Manby, venting his frustration by punching a drinks table.
Elsewhere, Brendan Dolan edged past Tavis Dudeney 3-1, highlighted by a 170 checkout, while Mensur Suljovic overcame Canada's David Cameron by the same scoreline.