The Spurs manager was speaking after his side secured a 2-1 victory at Aston Villa, a result that lifted them into 17th place and one point clear of the bottom three with three games remaining.
De Zerbi praised the overall performance as much as the result, highlighting both the intensity without the ball and composure in possession.
"Very pleased for the performance. We won a great game against a great Aston Villa side," he said. "Without the ball we showed courage and with the ball great quality."
Gallagher was central to the win, scoring his first goal for the club since joining from Atlético Madrid in January with a long-range strike that set Tottenham on their way. Richarlison added a second before half-time, with Emiliano Buendía pulling one back late on.
De Zerbi reserved special praise for the England midfielder's influence across the pitch.
"When Gallagher plays like this, we play with twelve players because you can find him everywhere," he said. "Great player, great passion, great qualities."
The victory marked a second successive league win for Tottenham - their first since August - and continued a turnaround that has seen them move out of the relegation zone after being 18th just weeks ago.
De Zerbi said rebuilding confidence had been key since his arrival earlier this year.
"I think it was a mental problem," he said. "These two wins can help us play with more confidence and better mentality."
Despite the improvement, he warned his players against complacency ahead of the run-in.
"The season is not finished yet. We have to play another three very tough games," he said. "We can't forget what the situation was before. It was a very sad moment and these memories have to stay in our heads."
Aston Villa manager Unai Emery defended his decision to rotate his side ahead of their Europa League semi-final, despite the defeat.
"I believe in the players," Emery said. "When we win, nobody asks about changes. When we lose, you ask why."
Tottenham face Leeds next before further matches against Chelsea and Everton as they look to secure their Premier League status.
