Jake Hesketh handed Martin Hunter’s men the lead early on with his sixth of the season, before Rodriguez finished expertly just after the 30-minute mark to double Saints’ lead.
Josh Sims added a third shortly after half time, before a second-half brace from Rodriguez completed his hat-trick and sealed an emphatic win for Saints.
There were two changes from the win at Middlesbrough on Monday, as full-backs Matt Targett and Yan Valery came in for captain Dom Gape, who missed out completely, and Jake Vokins who dropped to the bench.
Fresh from 60 minutes in that win against Boro, Rodriguez also started as he stepped up his recovery from a long injury lay-off.
The scoring was opened in just the sixth minute, as Sims won back possession in a dangerous area and crossed for Hesketh, who applied a calm finish, curling the ball into the bottom corner.
Experienced left-back Jose Enrique had the first real chance of the game for the hosts, as he flashed a volley from distance just wide of Paulo Gazzaniga’s goal.
After scoring the first, Hesketh turned provider to help Southampton double their lead. His pass released Rodriguez, who made no mistake as he slotted low beyond Adam Bogdan.
Gazzaniga had to be alert just before half time to maintain Saints two-goal advantage, as he beat away Ryan Kent’s low strike before Enrique headed over from the rebound.
Moments after the restart, Saints extended their lead. Valery’s cross was only cleared as far as Sims, who expertly looped the ball over Bogdan and into the far corner of the net.
Liverpool’s main threat was being provided by Kent, but again the attacking midfielder was denied by Gazzaniga, as the goalkeeper pushed away his near-post effort.
And Saints further extended their lead eight minutes later, as Rodriguez was released by Sims, and despite the best efforts of Bogdan, his powerful strike found the back of the net.
The striker soon completed his hat-trick, as he pounced inside the penalty area to divert a low strike from Armani Little beyond Bogdan.
The hosts rarely threatened to mount a comeback, as Hunter’s men held on to make it a double celebration for Saints against Liverpool following the first team’s win at St Mary’s.
Liverpool Under-21s: Adam Bogdan, Connor Randall, Tiago Ilori, Joe Maguire, Jose Enrique, Joao Teixeira (Sam Hart 68), Pedro Chirivella, Cameron Brannagan (c), Jordan Ibe (Ryan McLaughlin 70), Ryan Kent, Toni Gomes (Alex O’Hanlon 68).
Unused substitutes: Andy Firth (GK), Madger Gomes.
Southampton Under-21s: Paulo Gazzaniga, Yan Valery, Ollie Cook, Alfie Jones, Matt Targett, Harrison Reed (c), Armani Little, Sam Gallagher (Jake Vokins 88), Jake Hesketh (Ben Rowthorn 85), Josh Sims, Jay Rodriguez (Olufela Olomola 78)
Unused substitutes: Harry Lewis (GK), Richard Bakary.
Goals: Hesketh (5), Rodriguez (35, 55 & 64), Sims (47)
Referee: Richard Wild