Day birdied the first extra hole to scrape through after Henley birdied the 18th hole to take the match into extra time.
Day trailed through 11 holes but then won the 12th, 13th and 14th to move two ahead before the Henley fightback. A double bogey at the 15th cost Day that hole then the birdie at the 18th saw the 2012 Sony Open winner, Henley, keep the match alive.
Day now advances to round three where he will face Bubba Watson who survived a marathon against Jim Furyk, their match going 22 holes before Watson prevailed.
Australia's only other survivor from round one, Marcus Fraser, lost his second round match to Sweden's Fredrik Jacobson after losing the first two holes and never really recovering. Birdies at the 14th and 15th by Jacobson put the seal on the match, the final margin being 4&3.
Fraser earned US$95,000 while for Day he is guaranteed a cheque for US$140,000 even if he loses against Watson.