For the second game running, United dominated the play and had most of the chances, with Paul Pogba putting them ahead with his first goal of the season.
But just as against Middlesbrough in the FA Cup, the visitors were made to pay for their profligacy.
On this occasion, they had two first-half goals disallowed, while Nick Pope made excellent saves to deny Marcus Rashford, Edinson Cavani and Raphael Varane.
In contrast, Burnley made the most of their brief spell in the ascendency at the start of the second period, as Rodriguez ran onto Wout Weghorst’s through-ball to score his 100th career league goal.
It means United drop to fifth, with the three teams below them all capable of catching them if they win their games in hand.
Burnley are four points from safety, but are three unbeaten, even if their run without a win now extends to 10 matches.
Pogba’s sublime finish from Luke Shaw’s cut-back in the 18th minute was a thing of beauty.
Given far too much space inside the Burnley penalty area, Pogba astutely steered his first-time shot into the roof of the net, giving goalkeeper Pope no chance.
It was one of those moments which underline Pogba’s status as one of the best midfielders in the world.
But there is another side to the £90m man. Taking possession inside his own half close to the touchline, he spun round, away from a Burnley defender into space and was perfectly placed to switch play across the pitch.
Instead, Pogba smashed the ball down the touchline to no-one. Those infuriating moments have happened far too frequently since he returned to Old Trafford for a second time in 2016 and explains why not many United fans will shed a tear if Pogba leaves the club for nothing for a second time when his contract expires at the end of the season.
When United were chasing the game, Pogba didn’t really step up either, although for the second successive game, interim boss Ralf Rangnick will wonder about a VAR decision that went against his team.
This time, it was Varane who had what would have been his first goal for the club ruled out.
Even though the Frenchman was clearly onside as he ran onto a Bruno Fernandes free-kick before steering a header into the far corner, Harry Maguire was offside before he blocked Rodriguez’s attempt to get back ad defend. After consulting the monitor, referee Mike Dean ruled the effort out.