Llorente scored the only goal of the game in the 23rd minute, moments after Luis Suarez had a goal disallowed by VAR for offside.
However, Llorente seemed set to go from hero to zero for LaLiga leaders Atleti when he conceded a late penalty for handball.
Yet Atleti came away unscathed, Elche captain Fidel hitting the post with his spot-kick. Atleti have now not conceded from the previous three penalties they have faced.
The victory lifted Atleti five points clear in a four-way title tussle, albeit that gap will be closed should Real Madrid beat Osasuna in Saturday's late game.
With Barcelona having lost their game in hand on Thursday, Atleti would be crowned champions should they win all of their four remaining matches – including a huge game against the Blaugrana at Camp Nou next week.
"It was an obligation to win, it was very important for everyone," Llorente told Movistar.
"The team knew how to carry out the game. All the rivals are at the top and Elche are a great team. We take all three points and it gives us great satisfaction."
Asked if luck was on Atleti's side, Llorente replied: "Starting from the fact that I don't believe in luck, in the end, if [the penalty] got to the goal, [Jan] Oblak had guessed the right way.
"We got the points and it gives us a lot of strength for what remains.
"Winning is always positive, confidence increases, and today is one of those days."
It was a sentiment echoed by Simeone, who told a news conference: "It is time to work. The facts must be demonstrated on the pitch: that the team improves, plays with enthusiasm, is fierce and what I imagined and dreamed of when I arrived at Atletico."
Asked if Atleti must be considered favorites, Simeone said: "We are not in the moment to think; we are in the moment to do."
Simeone has now coached in 360 LaLiga games, surpassing Helenio Herrera as the Argentine to have managed the most matches in the competition.