A music video for the track directed by Sia herself appeared on YouTube on 12 November 2013. The video is not of a romantic nature. A power ballad, Hilary Hughes of The Village Voice named "Elastic Heart" a "grand, full electropop production," while Aimee Cliff from Fact characterised the song as "a pop song that bounces like rubber." Meanwhile, Rolling Stone reviewer Julianne Shepherd called it a trap song. Its lyrics address "the overwhelming strength [Sia] needed to convince herself that life was worth living after coming out of a crushing relationship." Having the chord progression of D–A–E–F♯m, the song is composed in the key of F# minor.
Blue Sullivan of Slant Magazine wrote that "'Elastic Heart' is a seamless and highly successful team-up that reads like a grimier after-market version of Lady Gaga's 'Do What U Want'."Spin magazine's Chris Martins stated that "Sia soared" and it was a "bubbling ballad," and Bradley Stern of MuuMuse claimed it was "a perfectly chaotic combination." "Elastic Heart" debuted at number 67 on the Australian ARIA Singles Charts on the chart issue dated 14 October 2013. The single also charted at number 27 on the Walloon Ultratip, and number 36 in Switzerland. The song fared better in New Zealand, peaking at number 7 on the national singles chart and was certified gold by the Recorded Music NZ (RMNZ). In the United Kingdom, "Elastic Heart" debuted at number 79 on the UK Singles Chart on 12 October 2013. On 17 January 2015, the single re-entered the chart at number 61.
In 2014, Sia recorded a solo version of "Elastic Heart" for her sixth studio album, 1000 Forms of Fear. It was released on 9 January 2015 by RCA as the fourth single from the album. Annie Zaleski from The AV Club labelled "Elastic Heart" a "striking power ballad," while AllMusic's Heather Phares picked the song as one of the three standouts from the album, alongside "Chandelier" and "Eye of the Needle". Aimee Cliff from Fact named it a "great example of how only Sia can truly rock a song written for (rather than by) Sia."
A still from the video featuring Shia LaBeouf and Maddie Ziegler, which sparked accusations of pedophilia.
On 7 January 2015, the music video for the song was released. Directed by Sia and Daniel Askill and choreographed by Ryan Heffington, the video features Maddie Ziegler, who previously appeared in the video for Sia's single "Chandelier", and actor Shia LaBeouf. In the video, Ziegler and LaBeouf wear nude and dirt-smeared outfits. Justine Harman from Elle likened the concept of the video to the plot of Titanic. Jason Lipshutz from Billboard summarised: "The entirety of the video features the surprising pair interpreting the song through various body contortions: they dance-fight, collapse in the middle of the cage, crawl toward and away from each other, and make some wildly fantastic facial expressions."
Upon its release, the video faced criticism due to the depiction of an adult and child dancing together, clad only in beige dancewear. Some commentators perceived the piece as pedophilic, and Sia later apologised on Twitter: "All I can say is Maddie and Shia are two of the only actors I felt could play these two warring 'Sia' self states. I apologize to those who feel triggered by 'Elastic Heart'. My intention was to create some emotional content, not to upset anybody."
The video was nominated for the 2015 VMA Award for Best Female Video. In June 2015, Billboard selected the video as one "of the 10 best music videos of 2015 (so far)".PopCrush also chose it as one of its favorites of the first half of 2015, commenting that the video's "lopsided choreography and filthy warfare yields a raw, junkyard beauty that doesn’t ask who will make it out alive, but whether escape is ever really on the table." As of September 2015, the video had received more than 406 million views on YouTube.
On 17 January 2015, Sia performed "Elastic Heart" on Saturday Night Live. For the performance, Sia sang the song with a short black veil covering the top half of her face, while Ziegler recreated the dance routine in the music video wearing a nude leotard and blonde wig. Later that month, Sia made a live rendition of the song on The Ellen DeGeneres Show. She also performed the song on The Voice UK on 28 March and The Voice US on 7 April 2015.