The Uruguayan set up Neymar and Jordi Alba to score the second half goals that gave Barca three vital points, reports Xinhua.
Barcelona coach Luis Enrique fielded a controversial starting 11, resting Suarez and Neymar from Wednesday’s Champions League trip to Ajax, while Adriano, Marc Bartra and young striker Munir also got a start.
Gerard Pique remained out of favour and started the game on the subs’ bench yet again.
The first half saw Barcelona create the best chances as Munir and Messi forced Almeria goalkeeper Ruben into a triple save after nine minutes.
After 37 minutes pressure on Messi saw him loose the ball and a smart pass from Jonathan Soriano set Thievy through on goal. The former Espanyol forward outpaced the Barca defenders and slotted the ball past the helpless Claudio Bravo to give Almeria a shock lead.
Luis Enrique wasted no time in making changes, bringing Suarez and Neymar for Pedro and Munir at the break, but the shape of the game hardly changed.
Barca equalized after 72 minutes after a magnificent turn and pass from Suarez left Neymar with just a tap in, although the Brazilian almost missed as his shot bounced agonizingly over the line off the back of a defender.
Jordi Alba gave Barca the lead in the 83rd minute as Suarez again turned provider with a cross from the right which the full back only had to prod over the line.