Whilst attending sixth-form in Croydon, a friend of La Havas', the singer and songwriter Allan Rose (who had attended the Brit School), introduced her to other musicians who would assist La Havas in the recording of her first demos. Through that same friend La Havas was also introduced to British singer Paloma Faith; she later sang backing vocals on tour for Faith. La Havas was co-writer and performer in The Paris Parade alongside Christian Pinchbeck (who designed the artwork for Lost & Found and is also now part of the duo Memphis Industries duo Elephant); their short-lived career would kick off La Havas' career in commercial music. In 2010, Lianne signed to Warner Bros. Records, spending two years developing her songwriting skills before releasing any music publicly.
Her first EPLost & Found was released on 21 October 2011 on the Labour of Love label, featuring Willy Mason on the opening track "No Room For Doubt". That same month, La Havas released the live EP Live From LA, which was made available for free download on her website. La Havas made her television debut on 21 October 2011 broadcast of BBC Two's Later… with Jools Holland, a programme that also featured Wisconsin folk band Bon Iver; soon after this on 25 October 2011 it was announced that La Havas would be the supporting act for Bon Iver's December 2011 North American tour.
Her official debut single "Lost & Found" was released in the UK on 30 April 2012, and her debut album, was released on 9 July 2012 on Warner Brothers. "Forget" was released as the second single from the album on 30 December 2011. "Lost & Found" was released as the third single from the album on 27 April 2012. "Is Your Love Big Enough?" was released as the fourth single from the album on 22 May 2012. The song was heavily contributed to the 2014 documentary film The Distortion of Sound, which features Lianne and her live performance of the single. "Age" was released as the fifth single from the album on 3 December 2012.
At the end of 2012, Is Your Love Big Enough? was named iTunes Album of the Year.
On 24 September 2012, September La Havas supported Alicia Keys at MTV 'Crashes' Manchester, a live performance in front of 1,000 people in Manchester Cathedral, which was broadcast in 164 countries. On 31 December 2012, she appeared on BBC Two's New Year's Eve show Jools' Annual Hootenanny singing Cow Cow Boogie. On 9 June 2013, La Havas played at the RockNess music festival in Inverness, Scotland, and on 30 June 2013 La Havas performed at the Glastonbury Festival 2013. La Havas also performed at the Isle of Wight Festival in June 2013.
At Metacritic, which assigns a normalized rating out of 100 to reviews from mainstream critics, the album received an average score of 75, based on 15 reviews, which indicates "Generally favorable reviews".Gaffa listed Is Your Love Big Enough? the 30th best album of 2012 by a non-Danish artist.