Taylor top-scored with 137, and Nicholls was unbeaten on 124 off just 80 balls, lifting the hosts to 364 for four after they lost the toss
With Sri Lanka chasing New Zealand's 319 for seven, they were on the ropes at 128 for seven when Perera opened up.
All-rounder Jimmy Neesham and opener Martin Guptill are also in the Twenty20 squad to face Sri Lanka
In reply to New Zealand's 371 for seven, Sri Lanka were all out for 326 with six balls remaining.
With de Grandhomme out and Corey Anderson injured, Jimmy Neesham and Doug Bracewell have been recalled as all-rounder options
The overwhelming victory gave New Zealand the series 1-0 after the first Test in Wellington was drawn.
At the crease were Dilruwan Perera on 22 and Suranga Lakmal on 16 with their 660-run target out of sight.
Play resumes on the fourth day Saturday with Dinesh Chandimal on 14, Kusal Menis on six and Sri Lanka needing another 636 runs
Relive Haris Rauf’s belligerent bowling in PSL-4
Shoaib Malik’s miserable end in ODIs
Umer Khan: The Giant Killer of Pakistan Super League
Babar overshadows Kohli at cricket's biggest stage
Glamorgan eyes Pakistan’s Fakhar Zaman for T20 Blast
The curious case of Aamer Yamin
Imam and Shaheen get their names on Lord's honors board
Amir determined to increase his pace
Stokes dismisses talk of redemption after World Cup victory
Asia Cup history and winners list