The Colombian's 35th minute goal pushed Wigan up two places to 14th and four points clear of the relegation zone with a first triumph over Liverpool after two draws and nine losses and after seven Premier League games without a win.
Liverpool remained sixth after its ninth loss of the campaign, a point behind fourth place Tottenham having played a game more.
"We didn't play well, especially in the first half. We were working hard without any game intelligence," said Liverpool manager Rafa Benitez. "We are unhappy. We have to show more character to be in the top four."
Last season Liverpool finished runner up to Manchester United, which tied its record of 18 league titles.
But Liverpool has not won the championship since 1990 and is struggling to compete with Tottenham, Manchester City and Aston Villa in the race for fourth place which would mean a shot at next season's Champions League.
Tottenham has 49 points from 28 games, Man City 49 from 27. Liverpool has 48 from 29 and, although it is three points ahead of Aston Villa in seventh, it has played three games more.
Although Fernando Torres hit the Wigan post with an angled shot early in the game, the home side played better and more threatening football on a surface newly relayed after the old one had been worn out by weekly use by both the town's football and rugby league teams.
With Liverpool lacking cohesion in midfield and frequently wasting possession, it was no surprise when Wigan went ahead but it took a Liverpool blunder to hand the home side the lead.
Dirk Kuyt gave the ball away to Emmerson Boyce deep in his own half and the Wigan midfielder advanced down the wing and then found Rodallega unmarked in front of goal to stroke it past goalkeeper Pepe Reina.
Glen Johnson made his first Liverpool appearance after a 12-game absence with a knee injury but a blunder by teammate Jamie Carragher almost handed Wigan a second goal in the 59th minute.
With Reina out of his goal, the defender's clearance hit Wigan striker Marcelo Moreno and looped towards goal only for Liverpool defender Sotiris Kyrgiakos to head off the line.
Liverpool almost equalised in the 76th minute, however, when Kuyt headed a cross down to Torres and the Spaniard's first time shot rolled past the far post.
