Rodriguez netted in the first half as the Clarets piled the misery on the their hosts, who have not won for eight league matches.
Eddie Howe's men, who stunned Nottingham Forest 5-1 in their previous match, secured back-to-back victories for the first time this season.
But Kenny Jackett's charges are facing a possible relegation battle after failing to score in a home league match for the third game running.
There was little sign of the struggle to come when Danny N'Guessan forced an early corner for the hosts, although James Henry's delivery failed to threaten.
Burnley responded soon after as Junior Stanislas made a determined run but was eventually thwarted by Scott Barron with the goal in his sights.
Millwall's Brian Howard set N'Guessan away down the right midway through the first half but he was pushed wide and his shot lacked accuracy.
Brian Easton and Andre Amougou's defensive efforts were denying the Lions any opportunities of note and their frustration was compounded in the 37th minute when Burnley opened the scoring.
Ross Wallace's corner was met by Rodriguez, whose contact was not perfect but good enough to steer past David Forde.
The half-time whistle was met with howls of disapproval from the home fans.
Jackett introduced John Marquis for Jay Simpson as the hosts went in search of an equaliser and their first home league goal since August 13.
Millwall showed far more purpose in the early stages of the second half, with Henry's threatening shot blocked away before Liam Feeney wasted the chance to tee up his forwards as he overhit a cross.
The crowd responded by getting behind their team and they thought they had something to celebrate when Marquis slid the ball into the net, but the linesman's flag was raised for offside.
Howard's curling free kick saw Lee Grant spring into action and he did well to keep his side ahead with a sprawling save.
Feeney showed excellent technique to fire a shot on target in the 70th minute but Grant was again equal to the challenge.
Referee Simon Hooper had a major call to make in the closing stages as Marquis went down in the box, but the substitute was booked for simulation.
Hamer Bouazza climbed off the bench to rifle a late effort narrowly over and Marquis saw his shot cleared off the line, but Millwall's poor run continued as Howe, the division's youngest manager, claimed victory over Jackett, the division's longest-serving boss.