Suarez, who had not scored since Nov. 7, gave Atletico the lead in the 41st minute as he got the slightest touch to turn a low cross from Kieran Trippier into the net.
Suarez then slid in to knock home from a delightful low pass by Yannick Carrasco in the 58th for his ninth goal of the season, taking him joint top of the scoring charts alongside Celta Vigo's Iago Aspas and Real Sociedad's Mikel Oyarzabal.
Elche got themselves back in the game with a header from Lucas Boye in the 64th following a well-worked corner routine but substitute Diego Costa sealed the points for Atletico by winning and converting a penalty in the 80th.
"I'm really happy to be back, to be able to train with my teammate," Costa said after the match. "I always say: 'life knocks me down but I get back up again,' because I have my family, my teammates, the doctors alongside me.
"When I'm on the pitch I try to give my all. It hasn't been easy, since I've been here things keep happening to me but I'm strong, I'll always fight to keep going. There are moments when you can't do anything, and things don't depend on you. I'm lucky to be at a great club.
"There are a lot of people out there who don't have a job and if something happens, it's really bad for them. I'm here with the best, the doctors, thank god for that. I always pick myself back up when something happens to me. I've always been like that, I won't change."
Asked about his partnership with Luis Suarez, he jokingly added "When I wasn't here, Suarez wasn't scoring, the b------... now I'm back and he scores twice. Let's hope he keeps going, he's a goalscorer, and the more goals he scores, the better for the team.
"We compete, with respect between us. That's what it's about, he started and scored, I came on and scored. If we want to do something big this year we have to do what the team that won La Liga did, that was a great team, whoever played, they were up to the task."
Costa also said Atletico can maintain their position at the top of the table and win the title for the first time since 2014.
"I'm a fighter, a winner, to get to Atleti it wasn't easy at all," he said. "If you can't dream big it's better not to live. We're a big team.
"We have everything to fight with the big teams and we have to be there."