Barcelona left it late but was able to keep 1 win on Sunday.
Real Madrid claimed a 1-0 win at Rayo Vallecano earlier on Sunday, placing additional pressure on Barcelona to pick up maximum points from its trip to the Vicente Calderon.
Alves got Barcelona off the mark in the 36th minute, but Falcao equalized for the home side just after the break. Messi then provided the match-winner nine minutes from time, sealing three points for the reigning champions.
Messi thought he had opened the scoring in the eighth minute following a penetrating run into the box, but the referee disallowed it as the Argentine handled the bouncing ball in the build up.
Barcelona eventually grabbed the lead nine minutes before halftime, and it was the talisman who started the move with a familiar run.
Messi effortlessly carved his way through the Atletico defense before unlocking Cesc Fabregas on the left side of the box. The former Arsenal captain slid a one-time pass in front of goal to find Alves, who calmly punched home his second La Liga goal of the year.
Falcao helped Atletico draw level with his 15th goal of the season, but Messi restored Barcelona's lead in rather controversial fashion.
The Argentine curled a free kick into the top corner in the 81st minute, but Atletico protested the shot, saying that the referee did not whistle for the restart. The goal stood, and Barcelona marched on to close Real Madrid's lead atop the table back down to 10 points.
Barcelona has won two straight in league play, snapping Atletico's seven-game unbeaten run in the process.
The Catalans sit second on 54 points, while Los Rojiblancos are tied with Sevilla on 32 points.
Real Madrid 1, Rayo Vallecano 0
Madrid, Spain - Cristiano Ronaldo scored the lone goal for Real Madrid in a 1-0 victory over Rayo Vallecano at the Teresa Rivero on Sunday, helping the La Liga leaders claim their ninth straight win in Spain's top flight.
The Portuguese star, who leads La Liga with 29 goals through 24 games, provided the breakthrough in the 54th minute thanks to a cheeky backheel from 10 yards out. His blind shot made it through traffic before bulging the back of the net, giving Real Madrid all three points.
It was an unfortunate day at the office for Rayo Vallecano as Piti struck the post and Michu was denied a clear goal-scoring opportunity by a brilliant tackle from Sergio Ramos in the first half. Michu was then sent off in the 89th minute for a reckless challenge from behind on Sami Khedira, capping a disappointing home match for Rayo.
Real Madrid continues to set a blistering pace in the league, taking a total of 64 points from a possible 72. The win also snaps a three-game winning run for Rayo Vallecano, which occupies 11th place on 31 points.
Sevilla 2, Valencia 1
Valencia, Spain - Sevilla handed Valencia its second successive league loss on Sunday, claiming an impressive 2-1 win at the Mestalla.
Tino Costa put Valencia in front in the 25th minute, but Gary Medel leveled the score before the break. Jesus Navas emerged with the game-winner in the second half.
With two wins in a row, Sevilla appears to be out of a rut that saw the club take just two points out of a possible 24. The Sevillistas improve to 32 points, while Valencia remains third in the league 40 points.
Villarreal 2, Athletic Bilbao 2
Villarreal, Spain - Villarreal and Athletic Bilbao played to a 2-2 draw in a thrilling match at El Madrigal on Sunday that saw three second-half goals scored in the span of six minutes.
Marcos Senna struck a wonderful free kick in the 10th minute to give Villarreal the lead, but Bilbao overturned the advantage in the second half thanks to a calm header from Fernando Llorente in the 62nd minute and a counter-attacking goal capped off by Markel Susaeta four minutes later. Nilmar, who came on as a second-half sub, quickly equalized in the 68th minute to see the home side earn a share of the points.
Villarreal, with 27 points on the year, goes three points clear of the relegation zone, while Bilbao moves into fifth place with 34 points.
Osasuna 2, Granada 1
Navarra, Spain - Osasuna edged Granada with a 2-1 win at the Reyno de Navarra on Sunday behind two first-half goals in the span of three minutes.
Javad Nekounam and Dejan Lekic bagged goals in the 26th and 29th minutes, respectively, to hand Osasuna all three points. Mikel Rico pulled a goal back in the second half, but it proved to be nothing more than a consolation strike.
Osasuna has won two of its last three league matches as it sits tied with Athletic Bilbao and Malaga on 34 points, while Granada occupies a spot just two places above the relegation zone with 28 points.
Real Sociedad 1, Mallorca 0
San Sebastian, Spain - Real Sociedad jumped ahead of Mallorca in the La Liga table after Imanol Agirretxe scored the lone goal in a 1-0 win at the Anoeta on Sunday.
Agirretxe lashed a shot into the upper corner 11 minutes from time, giving Sociedad 30 points for the year.
Mallorca sits four points above the relegation zone on 28 points.