Bolton 0 Liverpool 1: Torres the architect as Reds lifted by Rodriguez's strike
The Argentinian latched onto a brilliant backheel by Fernando Torres to settle a competitive encounter at the Reebok Stadium four minutes from time.
Both sides had created numerous chances in an evenly-matched contest but Liverpool left satisfied that their near month-long stay in the bottom three was over.
The match had looked like bringing more frustration for Liverpool and under-pressure manager Roy Hodgson as Sotirios Krygiakos twice went close and Torres squandered a good early opportunity.
The win drew them level on points with Bolton, who had been enjoying their best start for four years and were looking set for a seventh draw of the campaign.
They might even have won it themselves had a glancing header from Kevin Davies late on fallen the right side, for them, of a post.
Hodgson named the same Liverpool team for the third successive Premier League game as he attempted to instil the cohesion they have been lacking.
It initially appeared to work as the Reds, hoping to build on the previous weekend's morale-boosting win over Blackburn, started confidently and created two early chances for Torres.
The first came after three minutes as Steven Gerrard swung in a corner but Torres could not keep his header on target.
His next opening came just five minutes later but, despite netting the winner against Blackburn, the Spaniard still showed signs of the rustiness that has hampered him all season.
Torres raced through on goal from Gerrard's pass but delayed a shot and then mis-kicked as he attempted to stab past Jussi Jaaskelainen from close range.
Bolton's first attempt on goal came when Stuart Holden blasted a free-kick at the wall from 25 yards.
After that Bolton gave as good as they got and Jose Reina parried a vicious volley by Holden from distance after the US international was teed up by Davies.
The hosts appealed for a penalty as Lee Chung-yong cut inside from the left and went down under a challenge from Jamie Carragher but referee Martin Atkinson waved play on to few complaints.
Liverpool attacked but Joe Cole scooped a long-range effort well over the bar and Gerrard did likewise from a free-kick.
Fabrice Muamba shot from long range for Bolton but his effort curved well away from goal.
Torres set up Gerrard but the Reds captain drove wide from the edge of the box.
Just before the break Carragher survived another penalty appeal after the faintest of touches with the hand.
Bolton made the early running in the second half and twice forced Liverpool to clear unconvincingly without creating a clear-cut chance.
Gerrard tried to raise Liverpool's fading tempo with a powerful run and long ball for Torres, but a promising attack was ended as Muamba collided with Lucas.
Liverpool suffered a blow just after the hour as Cole was forced off and was replaced by David Ngog.
Krygiakos bundled into Davies but Atkinson deemed the Bolton captain to have gone down too easily.
Torres nodded back down for Rodriguez to attempt a spectacular overhead kick which went just inches over. Bolton replied as Johan Elmander, looking for his first home goal of the season after hitting five away, showed great control and drove just wide of the post.
Liverpool roused themselves late on and carved Bolton open with a delightful move to set up the winner for Rodriguez.
Torres was the creator as he nutmegged Gary Cahill with a clever backheel and Rodriguez raced through.
The Argentinian showed great composure as he clipped the ball in off the body of Jaaskelainen.
Cole could miss next Sunday's game against former club Chelsea with a hamstring injury.
The 28-year-old, who moved to Anfield on a free transfer in the summer, limped out at Bolton just after the hour.
The club do not feel the problem is serious but his ability to train this week is likely to be affected.
Manager Hodgson said: "He has a slight hamstring injury and we don't know the extent of it until after he has a scan."
Bolton Wanderers: Jaaskelainen, Steinsson, Knight, Cahill, Robinson, Lee (Moreno 74), Holden, Muamba, Taylor (Blake 88), Elmander (Klasnic 81), Kevin Davies. Unused substitutes: Bogdan, Mark Davies, Ricketts, Cohen.
Liverpool: Reina, Carragher, Kyrgiakos, Skrtel, Konchesky, Lucas, Meireles (Poulsen 89), Maxi, Gerrard, Cole (Ngog 61), Torres. Unused substitutes: Hansen, Jovanovic, Spearing, Shelvey, Kelly.
Referee: M Atkinson (W Yorkshire).