Spurs looked set to end their long wait for a league victory when Xavi Simons curled home a superb effort late in the second half to restore their lead. Earlier, Pedro Porro had opened the scoring before Kaoru Mitoma cancelled it out just before the break.
But deep into stoppage time, Georginio Rutter struck to snatch a 2-2 draw, extending Tottenham's winless league run to 15 matches and leaving them stuck in the relegation zone.
Despite the crushing nature of the result, De Zerbi insisted the performance offered genuine encouragement and even claimed his side are capable of winning all five of their remaining fixtures to avoid the drop.
"I always believe in the qualities of the players," he said. "In this moment we need spirit, attitude and mentality. We have five games, 15 points, and this team is able to win five in a row."
The Italian, still early in his tenure, made it clear that belief and mentality will be decisive in the run-in. He warned that negativity would not be tolerated, demanding players return to training with the same intensity and positivity shown on the pitch.
"I have no time to see negative people or sad players," he added. "They have to be focused and come with a smile, otherwise they go home."
Tottenham's latest setback leaves them a point from safety, with pressure mounting as rivals around them continue to pick up results. Yet De Zerbi believes the signs of improvement - particularly in attack - suggest his team can still escape trouble.
"I saw character, organisation and quality," he said. "We deserved to win against a great Brighton side. Now it's too easy to think negative, but we have to move on and prepare to win again."