The Clarets striker, who was handed professional terms at Turf Moor under the now Forest boss, condemned the East Midlands side to a sixth consecutive league defeat with a headed goal in each half.
Rodriguez only denied himself a hat-trick by misfiring from the penalty spot when Ben Mee fouled Garath McCleary 10 minutes after the interval.
Forest have now failed to score on their own turf since victory over Ipswich on November 19, and enter February second-bottom of the npower Championship league table.
Cotterill handed 18-year-old defender Jamaal Lescelles his first start for Forest in the wake of Wes Morgan's departure to Leicester, and installed Adlene Guedioura, signed on loan from Wolves on Monday, in midfield at the expense of Jonathan Greening.
Burnley made just a single change, Millwall loanee Josh McQuoid handed his full debut, to leave Ross Wallace consigned to the bench.
The away side threw down the gauntlet with only three minutes played, Rodriguez marching into the six-yard box to head home Kieran Trippier's pinpoint cross with relative ease.
It was the 22-year-old's 15th goal in all competitions this term.
Their assignment to avoid a sixth consecutive league defeat now confirmed, Forest responded brightly to the early setback in an otherwise uneventful half.
Ishmael Miller danced on to a promising through ball to give Lee Grant food for thought, but his dragged shot was cleared by David Edgar.
Guedioura ignited sparks of Premier League quality in the Reds midfield, combining well with Lewis McGugan and McCleary to harass the Clarets backline.
But for all their invention, the end product was sorely lacking, Guedioura clipping tamely into Grant's arms and fellow debutant Lascelles nodding wide from McGugan's corner, when from both positions it had been easier to score.
Wallace made an unexpected entrance on the half-hour mark as Keith Treacy was withdrawn after showing signs of discomfort on his raids down the left wing.
The diminutive Scot provided a new creative outlet for the visitors before the break, twice feeding Charlie Austin, who was denied any sight on goal by the close attention of the Forest defence.
A cagey start to the second half preceded a golden opportunity for the Clarets.
With 56 minutes on the clock, McCleary hustled into the box only to be felled by Mee. Rodriguez stepped up to take the resulting penalty, but instead skied high into the visiting crowd to present Forest some hope of entering the contest.
But it was not a gift, instead a sign of worse to come for Cotterill's men.
In what was almost a carbon copy of his opening salvo, Rodriguez was once again the recipient of a canny cross from Trippier, directing the ball beyond a flailing Lee Camp to double the advantage.
Deflated and disheartened, Forest failed to produce a meaningful response as time slipped away at the City Ground.
Eddie Howe's side, though, pushed on in search of a third goal to consolidate an eighth league victory in 11 outings, Marvin Bartley drilling agonisingly wide of the target as full-time approached.