Portsmouth were seconds away from being relegated today, but a late goal gave them a dramatic 4-3 win over Doncaster Rovers, sending their opponents down instead.
Although Appleton understands that his side face an uphill struggle to survive, he is refusing to give up just yet.
"Our chances of staying up are not great," he said. "Even if we win the three games, all Bristol City have to do is look after their own ship. I think if it was a Grand National we'd be a 500-1 shot.
"What today does do though is it gives me confidence we can win three games. There's an opportunity there and we will continue to fight as hard as we possibly can.
"I'm glad we've got two home games now. The fans will hopefully push and squeeze the ball into the back of the net for us. Who knows? We live to fight another day. Let's see where we are on Tuesday."
Pompey's three remaining matches come against Crystal Palace, Derby and Nottingham Forest.