"I spoke to the players prior to the game about belief and desire and I saw that in abundance today," Pressley told BBC Scotland.
"If we continue to show that desire then I've no fears about our survival."
A David Kucharski own goal brought Falkirk level at half-time, but Hearts built up a 3-1 lead before Pedro Moutinho's late consolation.
"It certainly was close and there are a great deal of positives to be taken from today's performance," said Pressley.
"We came here, we found ourselves 1-0 down after an early goal and battled back and showed that desire to get ourselves back in the game.
"We then started the second half very, very well and I thought we were unfortunate not to be leading.
"In the last 10 or 15 minutes, Hearts were certainly on the back foot and hanging on there.
"The football aspect is absolutely not in any doubt about these players here.
"We just need that wee bit of a winning mentality and I saw a lot today to suggest that'll come."
Source: BBC Sport