Troy Deeney headed a second-half own goal that secured three vital points and lifted Palace to 16th - four clear of the bottom three.
It also eased much of the pressure that had been building ahead of their next four games against Chelsea, Southampton, Arsenal and Leicester.
A wider total of four wins from six, as well as a second in succession at Selhurst Park - where Allardyce previously felt his team "feared" playing - has given them some momentum as they bid to preserve their Premier League status.
It has also taken them to within three points of 13th-placed Watford, and when asked if he felt they were on course for survival, Allardyce responded: "It's on. It's on.
"We've won three on the trot, four out of six; we've made the team better all round. It's nice to look at the team and say 'why change (personnel)?' Prior to this I'd been changing week-in, week-out almost to find the solution to get a result.
"We had to tough it out for long periods. In the end, we produced a bit of quality to win.
"The 1-0 is very sweet for us, and that comes by our outstanding defending. We got what we deserved by a quality ball from the free-kick.
"(We have) another two weeks' break, so you want to finish on a high again.
"They can relax with their families, build up their strength and resilience for the little game we've got next time round against Chelsea. We can look forward to that now, because of what we've done.
"We might have put it (the fear of playing at Selhurst) behind us somewhat, and that's down to new players coming in who haven't got that fear, because they haven't played here before.
"All four (Mamadou Sakho, Patrick van Aanholt, Jeffrey Schlupp and Luka Milivojevic) have made a big contribution. That's lifted the rest of the players who were here, and lifted their fears a little bit. The change in the team out of possession is one of the reasons we've turned this around."
Watford manager Walter Mazzarri complained Palace's goal should not have stood, because he felt Milivojevic had fouled M'Baye Niang in the build-up to the free-kick from which the hosts scored, but he defended Deeney's game-changing contribution.
"We were not lucky at all," said the 55-year-old Italian. "There was a foul on Niang just before the goal, which means that we were with 10 men when they scored. It was also at the moment when we were doing our best.
"In my opinion the game should have ended 0-0.
"It's an own goal; this can happen. He shouldn't apologise for this. I don't think he's happy, but it was an unlucky day: he ran and gave everything for the team so he shouldn't apologise, it was unlucky."