Suarez ensured he made the headlines for the right reasons after showing Fulham fans the middle finger on Monday, as he nodded home Charlie Adam's pinpoint cross for his first league goal since the October 1.
Liverpool dominated the game but were frustrated by Neil Warnock's Rangers with Czech keeper Radek Cerny pulling off a string of fine saves.
After four successive home draws, the Reds struggled in front of goal once again with Suarez and Rodriguez both missing good first half chances.
But the Uruguayan ? who was hit by an FA charge for his Craven Cottage gesture ? got on the end of Adam's cross after the break to ensure Liverpool would register all three points.
Suarez had a golden opportunity to open the scoring after just nine minutes but his free header was directed straight at Rangers keeper Radek Cerny after a fine cross from the right by Stewart Downing.
Dirk Kuyt then failed to find anyone with a low cross across the face of goal before Rodriguez played in Suarez who hit wide with the outside of his boot.
Just after the half-hour mark Kuyt played through Rodriguez whose low strike across goal was kept out by the fingertips of Cerny.
And Suarez was in again before half-time after a driving run from Downing but his close-range lob was saved by Cerny.
Having dominated at home but failed to beat the other promoted teams Norwich and Swansea, the home fans would be forgiven for wondering if the Reds were ever going to score.
But they did not have to wait long in the second half to see their side finally break the deadlock.
Charlie Adam found an unmarked Suarez from the right flank and the 24-year-old made up for his earlier missed header to nod home from six yards out.
It was Suarez's first goal since getting two in the Carling Cup in late October and his first in the Premier League since the Merseyside derby.
The situation worsened for Rangers when Anton Ferdinand was forced off with a hamstring injury.
And Liverpool nearly had a second on the hour but Cerny produced a magnificent save to keep out Rodriguez's pointblank strike after lovely link-up with his fellow South American Suarez.
Cerny was keeping the Hoops in the game and had to tip over Kuyt's shot which he struck into the turf.
QPR came close to nicking an equaliser when Danny Gabbidon diverted over Joey Barton's inswinging free-kick.
Substitute Craig Bellamy curled a free-kick into the side-netting but the visitors had a late penalty shout when Alejandro Faurlin went down under Martin Skrtel's challenge.
Bellamy's cross was deflected onto the bar by Shaun Wright-Phillips in the dying seconds of the game but one goal proved enough for Dalglish's side who move up to sixth in the league.
Liverpool: Reina, Johnson, Skrtel, Agger, Jose Enrique, Maxi (Bellamy 78), Adam, Henderson, Downing (Shelvey 86), Suarez, Kuyt. Subs not used: Doni, Carroll, Coates, Carragher, Kelly. Booked: Agger, Bellamy. Goals: Suarez 47.
QPR: Cerny, Young, Gabbidon, Ferdinand (Orr 50), Traore, Mackie (Hill 78), Barton, Faurlin, Wright-Phillips, Smith (Campbell 66), Bothroyd. Subs not used: Putnins, Derry, Buzsaky, Connolly.