The 1-0 win at Ashton Gate lifted PNE up to 18th place, having gone into the weekend fourth bottom.
Results went their way with late goals denying Millwall a point at Norwich City and Hull a victory at Birmingham.
There is a two-point gap between Neil’s men and the bottom three but they have squeezed up on the three teams above them who were all beaten over the weekend.
North End manager Neil said: “The priority had to be to win at Bristol City.
“Had we lost and other teams had won, we could have ended up slipping into a zone where we don’t want to be.
“Now that we have won, we have gone above two or three teams.
“We have got two or three teams in our sights who are only a couple of points above us. I think the situation we are looking at has become a bit better.
“We have also managed to create a wee gap between us and teams at the bottom.
“Things can quickly turn and we have given ourselves a bit of respite in terms of what is below us.
Now we have to make sure we really attack the next batch of matches, get ourselves into the middle of that table and start looking up.”
Three of North End’s four league victories this season have featured clean sheets, as did the win in the League Cup over Leeds.
Their defending has seen a gradual improvement in the last few weeks.
After shipping 17 goals in six league games through September and early October, Preston have tightened up at the back.
They have still let in six goals in the last six games but three of those came in the 4-3 victory over Brentford.
PNE restricted the Robins to one shot on target at the weekend, that a free-kick from Josh Brownhill which Declan Rudd tipped over the bar.
The clean sheet got some recognition from the Football League, who had Ben Davies in their Championship team of the weekend.
Said Neil: “I thought we were as comfortable as we have been this season.
“The guys were heading and clearing balls out of the box, making good decisions.
“To start with we didn’t deal too well with the first long ball up to the big striker but we got to grips with that as we went along.”