Andres Iniesta sprinted out of defence to help set up the opener on 24 minutes. The Spain international found Thierry Henry in space on the left and Eto'o turned his precise centre into the net past goalkeeper Inaki Lafuente.
The home side were 2-0 ahead before the break when Lionel Messi cut in from the right wing and threaded a pass through to Eto'o who turned defender Gerard Autet and rifled home.
Alves burst into the area and smashed in from an acute angle with just over an hour gone before Sporting midfielder Kike Mateo pulled one back two minutes later with a smart strike from the edge of the area.
Barca coach Pep Guardiola said he was impressed with his players' ability to dominate possession and swiftly turn defence into attack and paid tribute to the two goal scorers.
"Alves never tires and is an extraordinary player and Eto'o helps us a great deal with his goals and his sacrifice," he said at a news conference.
PERFECT START
In Sunday's earlier matches, Sergio Aguero scored one and twice set up Diego Forlan to help seal a 3-0 win for Atletico Madrid at Recreativo Huelva and give new coach Abel Resino the perfect start in his first match in charge.
Aguero stole in among the Recre defenders to nod Florent Sinama Pongolle's cross from the right high past Recre goalkeeper Asier Riesgo in the third minute.
Two sliderule passes from the Argentina international then set up strike-partner Forlan to score a couple of well taken goals from tight angles to put the visitors 3-0 up by halftime.
Atletico's victory, their first this year, comes after Resino replaced sacked Mexican Javier Aguirre last week and moved them up to sixth with 35 points from 22 matches.
"I could see the players were very committed and concentrating well," Resino said at a news conference. "It was a good contribution towards the team's recuperation."
In the late match, fourth-placed Valencia missed the chance to climb above Sevilla into third when they lost 1-0 at Osasuna in slippery conditions at the Reyno de Navarra after Saturday's heavy snowfall.
Patxi Punal's freekick in the 57th minute evaded the Valencia defenders and Miguel Flano sent a spectacular diving header flying past Luis Cesar Sanchez into the net.
The win moved the Pamplona-based club out of the relegation zone into 17th above Numancia, Espanyol and Real Mallorca.
Sevilla lost further ground when they went down 2-1 to city rivals Real Betis on Saturday, a third straight defeat that leaves them on 38 points.
Villarreal stay in fifth, level on 37 with Valencia, after Italian striker Giuseppe Rossi scored twice to give them a 2-1 home wi
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