'Snakebite' Wright came through a routine-looking victory on paper at Alexandra Palace, but did not score particularly well as he recorded an average of just 83.51.
But some clinical finishing on the doubles carried his display, allowing him to brush aside Dutchwoman Van Leuven.
"It's just great to be back out here," Wright told Sky Sports. "I know I didn't play well, but I know I will in the next round.
"You've got to grow into the tournament. I mustn't show them too much for the next round!
"I'll be OK next round."
Wright broke Van Leuven's throw from the get-go, then slapped a mammoth 152 checkout to ensure he consolidated.
The scoring from both players left a lot to be desired, but Wright went a perfect three for three on the doubles to seize a 1-0 lead.
Set two remained on throw and Van Leuven notched a ton-plus out of 108, but the frame would go Wright's way under pressure on double 19.
Some poor doubling from Van Leuven allowed Wright to break open the third and that was all that was needed as he raced away to earn a straight-sets win.
Menzies rages in Manby meltdown
Frustration boiled over for the 26th seed Menzies as he landed several uppercuts on the drinks table following a stunning 3-2 loss to debutant Charlie Manby.
After losing the deciding set to the 20-year-old, Menzies started unloading on the underside of the table, which sent a water bottle and equipment flying.
The Scot was seen apologising to the Ally Pally crowd as he walked off stage, with his right hand appearing to be bloodied with a large gash.
"I think (Menzies) will regret that for the rest of his life," former Premier League champion Glen Durrant said during commentary for Sky Sports. "It was not a good watch."
Menzies started the match strongly with seven perfect darts in a winning first set, before Manby responded with a 115 checkout in the second to level.
They exchanged sets again to force a decider where Manby hit the Shanghai 120 finish to break Menzies' throw, then won the contest needing seven match darts.
Meanwhile, Brendan Dolan cracked a 170 checkout but had to contend with nine 180s from Tavis Dudeney in an entertaining 3-1 win to open the session.
Northern Ireland's Dolan reeled in the Big Fish to close out the second set in emphatic fashion after finishes of 106 and 101 earlier in the match.
Austria's Mensur Suljovic took care of Canada's David Cameron 3-1.
Monday's 2026 PDC World Darts Championship results
Afternoon session
Brendan Dolan 3-1 Tavis Dudeney
Cameron Menzies 3-2 Charlie Manby
Mensur Suljovic 3-1 David Cameron
Peter Wright 3-0 Noa-Lynn van Leuven
