Johnson prodded the hosts in front from a corner before Kuyt added a second from the spot after Danny Gabbidon was penalised for an obvious handball. Rodriguez headed in a third for Roy Hodgson's side to put the game well beyond the east Londoners, who have now won just once in 14 Premier League matches this season and are rooted to the bottom of the table.
The visitors have not won at Anfield since 1963 and supporters called for manager Avram Grant to be sacked as their team slipped to another miserable defeat. Liverpool, who saw a six-match unbeaten run ended with a 2-0 defeat at Stoke City last weekend, will now go into next Sunday's trip to Tottenham Hotspur with renewed confidence.
Johnson marked his return to action with a quick response to the criticism he received from Hodgson last week. The England defender was picked out by a Kuyt pass and broke into the area, staying on his feet despite a strong challenge from Herita Ilunga and sending a cross to the far post that was headed to safety by Lars Jacobsen.
The pressure from the hosts continued with David Ngog cutting in from the left and hitting a low shot that was turned away by goalkeeper Robert Green. Argentina winger Rodriguez then missed with a powerful long-range drive with Liverpool keen to make a purposeful start.
Ngog continued to impress and withstood a foul from Gabbidon to slip a pass to Fernando Torres, whose shot flew wide of goal. West Ham were continually cut apart and when Kuyt rolled Ngog into space and the Frenchman crossed, Torres was only denied by a block from Radoslav Kovac. Then Green palmed over the bar from a drive from Raul Meireles.
But from the corner that followed, Johnson controlled on his chest and then fired past Green with a low shot for his first goal of the season, 18 minutes in. Soon the advantage was doubled following more dreadful defending from West Ham.
Nobody even attempted to clear Dutch forward Kuyt's bobbling cross, which ran all the way through to Torres. The Spain striker attempted to lift the ball into the West Ham area only for Gabbidon to handle twice. Even though Kuyt's 27th-minute penalty was far from convincing, Green dived the wrong way to take Liverpool two goals in front.
Carlton Cole finally threatened for the visitors, heading straight at goalkeeper Pepe Reina from a Luis Boa Morte cross. But soon Liverpool made it three after Green made a low save from Torres. West Ham failed to clear and Paul Konchesky's curling cross was glanced into the corner of the net by Rodriguez 38 minutes in.
Mark Noble drilled wide from distance as the visitors attempted some sort of response. But Kovac pulled off another fine block from Ngog to keep the deficit at three goals. And then Torres failed to hit the target with another effort from just outside the area. Rodriguez then drove over the bar as the Reds continued to create openings whenever they tried.
Reina had an anxious moment when he was forced to tip over his bar from Frederic Piquionne. Rodriguez just missed the far corner after breaking down the left and clipping an effort across goal and substitute Fabio Aurelio was also close with a powerful shot from 25 yards.
Hodgson was quick to deny speculation surrounding Johnson and praised the contribution of Christian Poulsen, who has struggled for form since arriving from Juventus in the summer.
He said: "Glen profited from the fact he has been obliged to undergo an enforced rest with the injury. It's taken him a while to get over the injury but he's been fit all week and he's trained hard all week and he was full of running.
"I had better get people to start a little bit more mischief, with Glen and other players if I can get better performances out of them. I'm not interested in storms in a teacup. Glen Johnson is a top class player, we all know that, and it was nice for the fans to see how well he can play."
He added: "Christian's a good player and I've felt sorry for him. If you can pay a price for being too honest, I think he has done that. In one or two of the games when the team wasn't playing well and he tried to do a little bit too much and failed and in a performance that was a failure he was deemed a failure. But he's far from that, he's a good footballer and he's shown what a good player he is."