The Islanders returned to the bottom of La Liga last weekend after a defeat away to Real Zaragoza and became the only team to lose to Manuel Jimenez's side in 2013.
On Sunday, they host Levante, who have lost their past four league games, and Manzano believes a victory for his side would change everything.
He told reporters: "There have been many turnarounds in the history of football and in the history of Mallorca, and why shouldn't we think that with a victory over Levante we'll be much closer to safety than right now?"
The coach admitted that his side's situation means they have no other choice than to beat the Valencians.
"On Sunday we have to produce positive pressure. We have our backs to the wall and we have to come up with the goods," he said.
"Pressure doesn't have to be looked at negatively, it can be a positive thing.
"We are going out to win the game. Right now our situation leaves us with no choice other than to go out and play to win, that's the mentality we should have - to win because we can win and because we should win."
Levante are currently being investigated by the Spanish Football League after their 0-4 defeat to Deportivo three weeks ago aroused suspicion of match fixing.
Manzano did not comment on whether he thought either side had been involved in any malpractice, saying "I don't know, everyone has their own conscience" but was certain "Levante will come here to win, because professionals always want to win".
He added: "I'm more concerned that my players do a good job."
Levante coach Juan Ignacio Martinez emphasised that he believes there was "nothing strange" about the defeat to Deportivo but admitted there was a lot of tension within his squad after the match, in which midfielder Jose Javier Barkero accused four of his team-mates of not showing enough commitment during the game.
"There was tension three weeks ago, the three days after the game were bad. When a team-mate speaks about things like that, it's bound to create a hostile atmosphere," he said.
Levante, on 40 points and nine clear of the relegation zone are still not guaranteed safety, and Martinez hopes his team can put any concerns about relegation to bed with a win in Mallorca.
"It's a special game for us because we need to take just one more step to secure survival," he said.