Ernesto Valverde's men completed a league and cup double after thrashing Sevilla 5-0 in last week's Copa del Rey final, regaining the Spanish crown from Real Madrid and relegating Deportivo in the process.
Barca knew they just had to extend their record unbeaten streak to 41 La Liga matches to seal a seventh title in 10 seasons, and Philippe Coutinho fired them into an early lead.
Messi's 38th-minute volley made it two, but Lucas Perez pulled one back for Deportivo before half-time and Turkish winger Emre Colak stunned Barca with an equalizer midway through the second period.
Barca finally saw off their stubborn hosts, though, as Messi pounced after a brilliant one-two with Luis Suarez before slotting in to complete his 30th La Liga treble.
The Catalan giants now hold an unassailable 11-point lead over second-placed Atletico Madrid, who have just three games to play, and are closing in on becoming the first Spanish side to go through a top-flight season without tasting defeat.
Andres Iniesta was surprisingly left on the bench after the midfield great announced earlier this week that he will leave the club at the end of the season.
But it was his expected long-term successor Coutinho who broke the deadlock in the seventh minute, bending an exquisite finish into the top corner for his third La Liga goal for the club since signing from Liverpool in January.
Deportivo had to win to stave off relegation for another week, but Clarence Seedorf's men were denied an equaliser when Perez was flagged offside after tapping in from close range.
Messi went close to doubling the advantage just before the half-hour mark, only for home goalkeeper Ruben to pull off an excellent one-handed save to turn his free-kick around the post.
- Deportivo cause problems -
Deportivo were causing Barca plenty of problems at the back, though, and centre-back Fabian Schaer should have levelled as he headed over when unmarked inside the six-yard box.
But Messi drilled a curling Suarez cross past Ruben at the keeper's near post to score his 30th league goal of the campaign, before on-loan Arsenal striker Perez swept home to halve the deficit just two minutes later.
Despite the close scoreline, Barca were playing as if the second half was a pure title party, with centre-back Gerard Pique even performing a few 'keepie uppies' on the left wing.
Their complacency cost them in the 64th minute, as Celso Borges unselfishly squared for Colak to score.
Suddenly the champions-elect were desperately trying to hang onto their unbeaten record, as goalkeeper Marc-Andre ter Stegen made a smart stop from another Schaer header.
But Barcelona showed their class as Suarez and Messi combined to terrific effect with eight minutes to play -- the latter clipping the ball in off the post.
Suarez again sent the Argentinian racing clear three minutes later, and the 30-year-old made no mistake to complete a hat-trick and a title-sealing victory.
Earlier on Sunday, Kevin Gameiro scored a second-half penalty just seven minutes after Fernando Torres had missed a spot-kick as Atletico Madrid claimed a 1-0 win at Alaves.
Diego Simeone's Atletico stayed in control of the race for second by moving four points clear of Real Madrid, who have a game in hand.
Fourth-placed Valencia's winless run was extended to four league games with a goalless home draw with Eibar, but their Champions League spot for next season will be confirmed if Real Betis fail to beat Malaga on Monday.
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