Nolan hat-trick helps hammer Sunderland
The Argentine midfielder latched on to a Fernando Torres backheel to secure a first away league win of the season for the 18-times champions who had again looked totally bereft of attacking ideas.
That was not the case at St James' Park where Newcastle United chalked up their biggest home win over local rivals Sunderland for more than half a century when they thrashed the 10-man Wearsiders 5-1 with a hat-trick from captain Kevin Nolan.
Liverpool had won their last seven games against Bolton but were desperately short of class at the Reebok Stadium.
They defended well, particularly in holding off the muscular presence of striker Kevin Davies, but created little in attack with Joe Cole having another miserable day and Torres also out of touch.
However, with a goalless stalemate looming, the Spain forward opened up the home defence with a deft backheel through his marker's legs to send Rodriguez through to poke the ball under Jussi Jaaskelainen.
Liverpool's second successive win lifted them from 18th to 12th on 12 points, 13 behind leaders Chelsea who won 2-1 at Blackburn Rovers on Saturday.
"It's nice to be out of that bottom three, we've been in there a while so it's been difficult to look at the table," Liverpool centre-back Jamie Carragher told Sky Sports.
"It was a tough game, not a great performance but we dug in."
Newcastle matched the four-goal margin of their 6-2 victory over Sunderland in 1956.
Although the visitors had Titus Bramble sent off after 53 minutes for a professional foul they were already 3-0 down.
Nolan put Newcastle two up with goals after 26 and 34 minutes and Shola Ameobi's penalty made it 3-0 at the break.
Ameobi added his second after 70 minutes after barnstorming striker Andy Carroll's header came back off the bar and Nolan nodded in the fifth five minutes later.
Darren Bent got Sunderland's goal in the dying minutes.
"It's fantastic, that hat-trick will live with me forever," said Nolan.
The morale-boosting victory came at the end of a week when Newcastle manager Chris Hughton had been subjected to media speculation that his job was on the line, forcing the club to issue a statement of support.
Hughton said: "We knew how big a game it was and you can see what it means to the fans. It was a fantastic team performance with some outstanding individual performances. We had that quality in front of goal today.
"It's been a tough week, but a very, very good day."
Sunderland boss Steve Bruce apologised to the fans on the club's website.
"We've let ourselves down and it couldn't have happened at a worse place," he said. "We were beaten well and truly, we were naive and didn't handle the occasion at all."
Aston Villa and Birmingham City drew 0-0 in the day's other derby.