Despite being bottom of the standings, Gustavo Quinteros's side went into the encounter knowing that victory could throw up the chance to move into the quarters as one of the best third-placed teams.
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Even winning would mean that Ecuador still need other results to go their way, but La Tricolor looked determined to keep their end of the bargain with a solid start and should have scored inside three minutes.
Bolanos would atone for that miss on 28 minutes by firing his country ahead after converting Enner Valencia's cut-back.
Miguel Herrera's side saw much more of the ball after falling behind but Mexico struggled to fashion the chances necessary to test Alexander Dominguez before the break.
Minutes after the break, Valencia could have doubled Ecuador's lead but his effort from eight yards flew over the crossbar - but, like Bolanos, the West Ham man redeemed himself.
He latched on to Bolanos's incisive through-ball before keeping a cool head to slot past Corona and boost Ecuador's hopes of securing a first win of this campaign.
Valencia and Bolanos wasted good opportunities to kill the game off once and for all toward the end, but Ecuador survived with the one-goal margin to at least put themselves in the hat for a last-eight spot.