João Palhinha headed home an 82nd-minute winner to give De Zerbi his first Premier League victory as Tottenham head coach and move the club to within two points of West Ham United in 17th place with four games remaining.
Speaking after the match, the Italian was direct about his confidence in staying up.
"I believe we will remain standing. If I stay here it's because I'm positive — not because I'm the coach of Tottenham. On the pitch it's not important if you're Tottenham.
"What matters are the qualities of the players and the human level of the players. West Ham is a good team, but so is Tottenham."
He also defended his squad's character under pressure.
"I work with them every day and these are professionals, good guys who are suffering from this situation. I just need to give them orders."
De Zerbi's task was complicated further by the loss of both Dominic Solanke and Xavi Simons to injury. Solanke hobbled off with a muscular problem in the 40th minute. Simons was stretchered from the pitch in the 63rd minute after what appeared to be a knee problem — an injury described in some reports as potentially serious.
De Zerbi gave an update after the game. "Solanke's injury is not a big problem. I don't know how many games we lose him. I would like to know more about Xavi though because the knee is always different to muscles. He felt pain but I spoke with him two minutes ago and he feels better than at the beginning."