[4] Speaking about the eponymous subject, guitarist Tom Morello remarked that "Cochise the Avenger, fearless and resolute, attacked everything in his path with an unbridled fury", adding that the song "kinda sounds like that".

[15] Due to the large number of pyrotechnics used in the video, local residents believed that the explosions were caused by terrorists; speaking about the filming process, Morello revealed that "The local police and news station literally received thousands of calls from people who thought the city was under siege ... All the freeways were blocked because they thought there was an attack occurring".

BBC Music's Amy McAuliffe noted that the song "sets the tone for the rest of the album [with] heavy, powerful riffage over rumbling bass lines and thumping colossal drums", comparing its style to that of Led Zeppelin and Black Sabbath.

[29] Blabbermouth.net's Don Kaye compared the style to Cornell's former band Soundgarden, highlighting the frontman's vocal performance and Morello's "meaty, straightforward" guitar riffs.

[30] Chris Heath of Dotmusic also praised the song's vocal and guitar performances,[31] while Paul McNamee of the NME claimed that while the album was "overblown, overlong, piss-poor", lead single "Cochise" is "fierce and furious".

[32] Chris Nettleton of Drowned in Sound praised many elements of the song, including Cornell's vocal performance and Morello's main guitar riff, but added that he wished the band "had chosen to break some new boundaries, instead of retreading an old town with new shoes".

[33] Entertainment Weekly's David Browne, on the other hand, proposed that the "elliptical lyrics" to "Cochise" feature a "hint of social consciousness".