The Italian came out fighting after Saturday's 1-0 home loss to Stoke, the Hammers' sixth straight league defeat, left them 17th in the Premier League, just one place above the relegation zone.
"I am determined to carry on," he told the London club's website.
"Losing on Saturday was emotional for everyone but I know we can turn it around and we have the ability to achieve our objectives.
"The owners have backed me and I am grateful for that. My only thought for now is to keep this club in the Premier League and that remains my goal. I will give everything I can to make this happen between now and the end of the season."
Zola, who joined the club in September 2008, has six games left to drag his faltering side away from danger, starting with a trip to Everton on Sunday.
"No one should write us off," he said. "Although we are in a serious position we have time to sort things out and be in control of our destiny at the end of the season."
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