The Canaries' 1-0 win at West Brom on Saturday ended their 11-game winless run and breathed fresh hope into their Barclays Premier League survival bid.
Newcastle's 1-1 draw at home to neighbours, and fellow strugglers, Sunderland on Sunday leaves Norwich two points clear of the drop zone.
Palace are just five points ahead of the Canaries after their own 11-match winless streak and Neil - whose side play Newcastle, the Eagles and Sunderland in their next three games - is ready to catch them.
"I took that one from where Norwich were two years ago. Five points above the relegation zone with seven games to play, sometimes you can feel as if it's going swimmingly and suddenly find yourself in trouble," he said.
"Trying to reverse that around is extremely difficult. Palace haven't won in 11 games now. They'll be concerned about that and we've got them to play. The more we drag in the better."
Robbie Brady struck in the second half to seal the points as Norwich kept their first away clean sheet of the season and Neil admitted his players knew it was now or never.
"You get yourself to a stage where if you don't start winning now it's going to be too late. We didn't pick up as many points as we'd like but you could see the urgency within their game and what it means to them," he said.
"That's the one benefit we've got over other teams about us. Our players have never turned against each other. We'll take strength from that."
Neil also held a team meeting away from City's training ground at a Norfolk hotel prior to their 2-2 draw with West Ham last month and feels it has paid off.
He added: "If I'm being honest they didn't disagree with anything so it made the meeting really easy. It's just about trying to get the players involved in it, making sure they have a clear idea of what we're doing."
Defeat ended West Brom's eight-match unbeaten home run and defender Jonas Olsson felt they blew a big chance to reach 40 points.
He said: "We have to get there first and hopefully once there we'll set new targets.
"It's very disappointing because we want to push higher up the table, there were three or four teams that are on 43 or 44 points that we wanted to get involved with."