Philippe Coutinho replied for Liverpool on the stroke of halftime, but Rodriguez made sure of victory in the 80th minute as Mauricio Pochettino’s side pulled seven points clear of the relegation places.
Queens Park Rangers were left seven points adrift of safety at the foot of the table after losing 3-2 to relegation rivals Aston Villa in a thrilling encounter at Villa Park.
After Jermaine Jenas gave QPR an early lead, Villa hit back through Gabriel Agbonlahor and Andreas Weimann, only for Andros Townsend’s deflected strike to draw the visitors level in the 73rd minute.
However, there was a further twist in the tale, as Christian Benteke stroked home the winning goal from Weimann’s cut-back with nine minutes remaining.
Benteke’s goal allowed Villa to put six points between themselves and the bottom three, although third-bottom Wigan Athletic have two games in hand.
In the day’s other game, Stoke City failed to reverse their disappointing recent form in a goalless draw at home with West Bromwich Albion.