The pieces feature a print called Black Lake, which Haapaniemi created after hearing the Icelandic singer and songwriter's description of the lava fields close to her house.
The pattern has also been used on a collection of accessories including scarves and purses, that will be available commercially.
"Initiated by a personal order from iconic artist Björk, Klaus designed a set of bespoke furniture inspired by [Björk's] own description of the Icelandic fantasy landscapes, that he then dreamt of bringing back inside the house," said the studio's co-founder Mia Wallenius.
The bespoke sofa with an angular timber frame features subtle hand engravings on its arms, back and feet, while chunky cushions are upholstered with the Black Lake print.
The woven tapestry is intended to be used as long rug that is draped over the sofa.
Inspired by the dark, steaming lake described by Björk, the circular carpet is decorated with unusual foliage, water lilies and mysterious lava stones.
While the furniture is custom-built by Finnish furniture studio Nikari, the tapestry is woven in a British textile mill in Bristol, and Nally, the circular volcanic lake-inspired carpet, is printed on wool in the Netherlands.
Before being sent to Björk's summerhouse, the Black Lake Collection will be shown publicly at Helsinki concept store and gallery Lokal.
The collection will be exhibited alongside other items made using the Black Lake print. The products, which include printed wool scarves and velvet clutches, will be sold exclusively at Lokal.
Lokal will also show part of Klaus Haapaniemi's Xenia etching series made with master printmaker studio, Himmelblau.
It is not the first time Björk has inspired a work of design, with Neri Oxman having previously produced a 3D-printed mask based on her musculoskeletal system.
Klaus Haapaniemi & Co was founded in 2010 by Klaus Haapaniemi and Mia Wallenius. The London-based Finnish design and lifestyle brand has since become known for its richly detailed, folklore-inspired patterns.