Mohamed Salah became the third player in Liverpool history to score 250 goals when he struck in first-half injury time, and Ryan Gravenberch scored in the second half after missing three games with an ankle injury in only their second win in eight matches across all competitions.
"The margins are small every single game - today after five minutes they (Morgan Rogers) hit the post," Slot told BBC Match of the Day. "We didn't concede a set piece, which was helpful, but also we got a bit lucky, which hasn't happened recently. Our second goal was a deflection."
"When (Villa) hit the post, that was their biggest chance. The last few weeks, every time that ball went in, now it didn't."
Slot added that being ahead allowed Liverpool to dictate play.
"The good thing about being 1-0 or 2-0 up is you can control the game a bit better with more possession. If you are down, you have to chase and take risks," he said.
Captain Virgil van Dijk echoed the sense of relief but warned against complacency, noting the impact of criticism during the slump.
"There's a lot of noise that you have no control over, and we have to deal with that as a team," he said. "Some of the takes are ridiculous, but you deal with it. We stick together."
The Dutch defender urged focus.
"Now it's time to start working, never get too high or too low," van Dijk said. "We live in a world where everyone can have their own opinions on so many platforms, and they know better. We have to stay away from that and focus on the hard work we are doing."
"Last season, we did some negative stuff, but everything was all sunshine and rainbows everywhere. Now it sounds like we're going to be in a relegation battle - that's how it works in the world."