The Uruguay international opened the scoring at Campo de Futbol de Vallecas but goals either side of half-time for Jose Angel Pozo and Alvaro Garcia Rivera put the hosts in control of the encounter.
Barca ultimately came from behind to claim the three points and remain top of the league thanks to Ousmane Dembele 87th-minute equaliser and Suarez’s winning strike on the cusp on injury time.
Yet the 31-year-old confessed that his side’s overconfidence was very nearly their downfall, acknowledging that manager Valverde had every right to be mad following their performance.
“We relaxed a little too much,” Suarez told reporters following his key brace. “I understand that Valverde may be angry. Matches like these are where games are lost.
“We relaxed a little and we'll have to analyse that, but the important thing was that we won the match.”
However, Valverde was far from angry with his forward, who also starred in Barca's 5-1 humiliation of Real Madrid last weekend that led to Julen Lopetegui's sacking as head coach.
"Suarez has been playing at the top of his ability for a very long time," Valverde said in the post-match press conference on Saturday. "We need him.
"Right now, he's scoring goals, more goals than perhaps before. We need him and we're extremely happy with him."
Suarez’s double lifted him to the top of the division’s scoring charts too, but he remained keen to stress that his focus remain on Barca rather than any prospective individual glory.
“Today was one of those nights where you came out and scored goals to help the team,” he stated.
Valverde’s side sit top of Group B after three games with nine points and can secure their progression through to the round-of-16 with a win in Italy.