The tiny Valencia-based club, whose annual budget of around 20 million euros is dwarfed by Barca's approximate 500 million, took the lead through Jose Barkero in the 14th minute before Malaga goalkeeper Willy Caballero was shown a straight red card for handling outside his area.
His replacement Ruben saved the resulting free-kick but Juanlu, one of a slew of Levante players over 30 years old, was quickest to react and poked the loose ball in to double the home side's lead with just over half-an-hour gone.
A Ruben howler gifted Levante a third in the 41st minute when he failed to clear and Arouna Kone, who scored the only goal in Levante's shock 1-0 win over Real Madrid last month, broke clear to tap the ball into an empty net.
Levante's fifth successive victory put them level on 17 points with Barca, who are top on goal difference after seven matches.
Ambitious and free-spending Malaga, owned by a member of the Qatar royal family and with a budget of around 150 million euros, are sixth on 13 points.
Levante's coach Juan Ignacio Martinez told a news conference it was important to remain realistic about what the club could achieve this season and said avoiding relegation was still the main objective.
"We have to enjoy this but we know that the league is very long and we are bound to suffer accumulating points," he said.
"We will achieve our goal of 42 points [to avoid relegation] in April, although hopefully I am wrong about that," he added.
On Saturday, La Liga top scorer Lionel Messi netted his ninth and 10th goals of the campaign in a 3-0 win for Barca, bidding for a club record-equalling fourth title in a row, at home to Racing Santander.
Barca and Levante have a one-point lead over Real Madrid, who thumped visiting Real Betis 4-1 thanks to a third hat-trick in two weeks from Messi's Argentina team-mate Gonzalo Higuain.
Sevilla climbed to fourth after their 2-1 win at home to struggling Sporting Gijon in Sunday's late kick-off, Manu del Moral smashing a long-distance effort into the top corner in the 15th minute at the Sanchez Pizjuan and Martin Caceres nodding a second early in the second half.
Valencia, who drew 1-1 at Real Mallorca on Saturday, are fifth on 14 points.
In Sunday's earlier kick-offs, Real Zaragoza's Portugal striker Helder Postiga struck twice, including a brilliant volley, to earn a 2-0 home win over Real Sociedad.
Ivory Coast midfielder Ndri Romaric scored the only goal as Espanyol ended a run of two defeats with a 1-0 victory at promoted Rayo Vallecano.
Athletic Bilbao are looking for a second straight win after a poor start when they entertain Osasuna on Monday.
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