Born in Oslo, Norway in 1977, Kornstad took up the clarinet in grammar school and eventually turned to the saxophone and studies at the Trondheim Jazz Conservatory.  He co-founded the group Wibutee and, by the time he graduated, the group was embraced by a community of artists centered around the contemporary music club Blå. It was there that pianist Bugge Wesseltoft heard and signed him to his label in 1998, and four Wibutee albums followed: Newborn Thing (1999), Eight Domestic Challenges (2001), Playmachine (2004) and Sweet Mental (2006).

In parallel, Kornstad organized the Kornstad Trio. Consisting of two more classmates from Trondheim, renowned bassist Mats Eilertsen and drummer Paal Nilssen-Love, they made a big impression when they were invited by guitarist Pat Metheny to play with him in 2001 at the Molde International Jazz Festival.

Kornstad’s interest in collaboration led to more critically acclaimed creative ventures, including a pair of duo albums with pianist Håvard Wiik entitled Eight Tunes We Like (2005) and The Bad and the Beautiful (2006), the latter nominated for the Norwegian Grammy. He also later recorded three duo albums with bass player Ingebrigt Håker Flaten, mixing religious folk songs with free jazz.
His own solo effort was reflected in the album Single Engine (2007).  Kornstad was now established in his own right as one of Norway’s leading jazz musicians. Dagbladet wrote “the definitive transition from ‘promising’ to ‘mature and original.’” His second solo album was released in 2009. Dwell Time (Jazzland) is recorded in the Sofienberg Church by legendary engineer Jan Erik Kongshaug. Downbeat Magazine dubbed him “one of Norway’s most original and daring musicians.” The third solo outing Symphonies in My Head came in 2011.
On a visit to New York in 2009, he discovered opera and decided to take singing lessons with a retired dramatic soprano. This led him to apply and be admitted in 2011 to The Norwegian Academy of Opera, where he completed his master studies as operatic tenor in May 2014. At the Oslo Opera House he performed several roles as part of his education.

Kornstad’s new direction, a meeting of jazz and classical song, took the form of Kornstad Ensemble. Seamlessly blending opera arias, lieder and jazz, the group featured harmonium and harpsichord in addition to a rhythm section. Spencer Grady in Jazzwise wrote: “Håkon Kornstad and his Tenor Battle troupe have fashioned a triumphant Bartók-reversal, the successful contextualising of high-end tropes into the folk firmament.” 

Their studio album Tenor Battle was released in 2015. Aftenposten wrote: “His project is a musical enrichment for everyone who loves classical music as well as those who like it when musicians yearn for the unknown. There is something magical, almost timeless and yet innovative and challenging in the way they treat these songs”.

​In September 2013 Kornstad was invited by American jazz saxophonist Joshua Redman to perform solo and duo in London’s Wigmore Hall. Another collaboration is the pianist Ketil Bjørnstad, with the album A Passion for John Donne out on ECM and Sanger om tilhørighet on Grappa.

In April 2015, the concert at Sarajevo Jazz Festival was released digitally as the album Live in Sarajevo, the first album to fully feature Kornstad’s combination of jazz saxophone and operatic singing. 
In May 2015, Kornstad sang in the newly commissioned opera Adam & Eve – A divine comedy by composer Cecilie Ore. The opera had its world premiere at the Bergen Festspiele, and then continued to the Ultima Festival in Oslo in September 2015.

In January 2017 Kornstad did his first Symphonic concert with the Tenor Battle repertoire. In this recording for the Norwegian national TV with the Norwegian Radio Orchestra, he presented traditional and new arrangements of opera arias and art songs, featuring both Kornstad the singer, the saxophonist and his debut as orchestra arranger. He also debuted as composer with symphonic arrangements of four of the pieces from his 2011 album Symphonies in my head. The highlights were released on the album Kornstad + KORK Live on the label Grappa. In 2018 he took this project to Cologne and the WDR Funkhausorchester for a series of concerts and radio recordings.

In 2019—in addition to extensive activity with his trio and solo projects—Håkon Kornstad is starting a collaboration with the Norwegian Chamber Orchestra. Furthermore he will take his symphonic project to Sweden and the Stockholm Wind Symphonic—and he will be one of the main artists at the Kirsten Flagstad Festival in Hamar and the Risør Chamber Music Festival, in addition to the jazz circuit.

Since his debut, Kornstad has presented his music in a majority of the European jazz Festivals, as well as festivals and venues in USA,
Canada, Brazil, Japan, China, Malaysia, Singapore and S. Korea.