"We were unlucky because we created scoring chances," Messi said. "We did everything possible to win."
Elsewhere, striker Ikechukwu Uche scored his sixth goal in his last three league games to help Villarreal beat Almeria 2-0, while Getafe erased a two-goal deficit to earn a 2-2 draw at home with Real Sociedad.
Far from the side that Barcelona humiliated in August, Levante lived up to its reputation as one of the league's best defenses by packing its area and largely limiting the visitors to long strikes that either crashed into bodies or sailed harmlessly wide.
Joaquin Caparros' side claimed the early lead when Vyntra rose up over Pique to head Andreas Ivanschitz's corner kick down and beyond the diving goalkeeper Victor Valdes.
Pique made up for his slack marking by scoring an almost exact replica of the opener, jumping higher than a pair of defenders to redirect Xavi Hernandez's corner inside the far post.
Messi, in his fourth game back after a long injury layoff, worked a one-two passing combination in close quarters with Cesc Fabregas to manufacture Barcelona's best chance in the 59th, but goalkeeper Keylor Navas saved Messi's shot. Garcia was well positioned to block Xavi's follow-up attempt, and three minutes later the center back used his foot to turn away Messi's goal-bound strike.
Barcelona's frustrating night was summed up when Fabregas violently struck the dugout cover after coach Gerardo Martino replaced him with Sergio Roberto in the 73rd.
With Neymar and Andres Iniesta both unavailable due to injury, Martino turned to Cristian Tello and B-team forward Jean Dongou as late substitutes. But Navas palmed Tello's late attempt wide to protect the point.
"In our first meeting (with Barcelona) it felt like a tribute game to Neymar in his debut," Caparros said. "My team has changed a lot because you can't make a team in a day, not even in two or four months."
Atletico started brightly when David Villa rifled in a clearance by Sevilla goalkeeper Antonio "Beto" Bastos in the 18th at Vicente Calderon Stadium.
But Atletico didn't seriously threaten to extend its advantage and it was undone when defender Juanfran Torres pulled Sevilla striker Carlos Bacca down by his shirt in the area.
Ivan Rakitic fired the spot kick into the corner of the net, just out of the reach of goalie Thibaut Courtois for the 73rd-minute equalizer.
Sevilla finished with 10 men when Alberto Moreno was sent off with a straight red card for a low tackle in the 89th.
"The draw is a pity," Courtois said. "We knew Barcelona's result before the game. It's too bad. When you go up 1-0 you hope to keep a clean sheet and win the match. It's tough, but that's the way it is."
Uche's strike against lowly Almeria was his 12th league goal this season, tying him with Sociedad's Antoine Griezmann as the competition's third-leading scorers.
Villarreal provisionally moved into fourth place with its third straight victory. Athletic Bilbao can reclaim Spain's last Champions League spot when it plays Valladolid on Monday.
