Although Atletico – who had Diego Costa sent off in the first half – looked set to hang on to a draw, Barca's dominance ultimately paid off as Suarez and then Messi struck late in the day as they did against Villarreal on Wednesday to land what is surely a fatal blow in the race for title.
Barca edged a tense first half, which saw Costa given a straight red card in the 28th minute for berating referee Jesus Gil Manzano, though they were unable to turn their superiority into a lead.
Atletico weathered the storm for most of the second period as well, with Jan Oblak producing an inspired performance, but he was finally beaten twice in the last five minutes as Messi claimed a record-breaking 335th LaLiga victory.
After a cagey opening, a moment of Messi inspiration opened the Atletico defence for the first time in the 14th minute, but the post came to Oblak's rescue after Jordi Alba collected the lofted pass.
Atletico's task became even trickier less than a minute later when Costa was given his marching orders for dissent, seemingly fuming at not getting a free-kick after a tussle with Messi.
Barca were dominant in the second half, with Messi testing Oblak from distance in the 56th minute, before the goalkeeper made an even better stop to thwart Suarez when one-on-one.
Oblak was finally beaten five minutes from time, however, as Suarez's gorgeous curling effort from just outside the penalty area found the bottom-right corner.
And Barca's second followed soon after – Messi riding a challenge from Jose Gimenez in the box and tucking a disguised effort inside the right-hand post to spark noisy celebrations at Camp Nou.
Curtains for Atletico
While a draw probably would not have done them much good either, a defeat surely ends Atletico's dwindling hopes of challenging Barca in the title race.
Barca will be deserved victors when they are eventually – as now looks increasingly likely – crowned champions. Atletico have consistently been a dull team to watch this season, while Ernesto Valverde's men have been anything but.
Costa loses his cool
Until Costa's sending off, proceedings were even. But in a moment of frustration he said a little too much to the referee. From then on Atletico's chances were minimal, as Barca dominated.
