With relegation rivals St Mirren and Kilmarnock playing each other in Paisley, the Bairns had the chance to climb off the basement for the first time since February.
However, Marco Paixao's first-half header gave Hamilton a deserved interval lead and although the visitors had to withstand a second-half pounding, they held firm.
Falkirk play fellow strugglers St Mirren next week and Pressley, in a short and sharp post-match press conference, slammed the table as he insisted: "I will make this clear today. I'm going to make this statement. There is no way that while I'm the manager of this football team we will go under.
"The players know that and it is clear. We will be ready next week, absolutely ready next week for the battle and the fight.
"I wish Saturday was tomorrow and I want to make it clear - we will be ready for next week.
"I'm bitterly disappointed. The one thing that went against us today was time. We just ran out of time.
"If you add another 10 minutes on to that game I'm very confident that there would have been another outcome, no doubt about it.
"We were relentless in the second half, with our pressure, desire and appetite to win a football match."
Asked why he was so confident, Pressley's snapped: "Because I see that effort and I know what we are all about.
Source: PA
Source: PA