Liverpool have taken a liking to playing Albion in the Premiership having now scored 13 goals in their last three visits to The Hawthorns.
In contrast, Albion have never scored a Premiership goal against Liverpool and have failed to beat them since 1981.
So their fate was effectively sealed after just seven minutes when Robbie Fowler gave the Anfield men the lead with his second goal for the club since his return to Liverpool from Manchester City, following a smart piece of play by the impressive Alonso.
Albion's comeback hopes were then extinguished after 38 minutes when Djibril Cisse pounced to score his 13th goal of the season after a long Alonso throughball.
The win, Liverpool's fourth on the trot in the Premiership, kept alive their hopes of clinching second place. But for Albion, their prospects of avoiding the drop are becoming slimmer by the week.
Even before the game had kicked off they had suffered a double setback when relegation rivals Portsmouth beat Fulham and Birmingham City took a point off Chelsea.
Albion have now picked up just one point from their last seven games and it will need a repeat of last season's dramatic Great Escape if they are to preserve their Premiership status.
The only good news for Albion was the display of half-time substitute Zoltan Gera who marked his second substitute appearance after a long battle to recover from a pelvic injury with an impressive cameo.
But the writing had been on the wall for Albion after just seven minutes when Alonso fed Cisse who moved in from the right wing and cut inside full-back Paul Robinson to cross for Fowler.
The former England international striker looked to be offside but the assistant referee's flag stayed down and Fowler calmly slotted the ball home at the far post.
Albion huffed and puffed in a bid to find a way back into the game and Diomansy Kamara and Robinson tried their luck with weak long-range shots that were comfortably dealt with by goalkeeper Jose Reina.
But Liverpool always looked capable of scoring whenever they went forward although Cisse's strike owed as much to poor defensive play from Albion as to the vision of Alonso.
Alonso collected the ball from Reina and fired forward a 60-yard throughball that dissected the Albion central defenders Steve Watson and Curtis Davies and fell invitingly into the path of Cisse who had once again escaped the attentions of Robinson.
Cisse then calmly took the ball around goalkeeper Tomasz Kuszczak before firing into an empty net.
Albion manager Bryan Robson hauled off the ineffective Kamara at half-time and replaced him with Gera in a bid to try and find a way back into the game.
Gera certainly added more spark to Albion - twice going close to reducing the arrears within five minutes of the re-start when he fired just wide and then had a first time volley saved at the near post by Reina.
Albion were a transformed side with Gera on the right wing and he again stung the fingers of Reina in the 67th minute with a first-time volley from Robinson's cross.
But as far as they were concerned the damage had already been done as Liverpool strolled to victory.