Paul Pogba put the visitors ahead in a first-half that they absolutely dominated. However, a well-taken goal from Rodriguez was just rewards for a Burnley team that rallied well in the second 45 minutes.
Story of the match
For the first time in his management, Ralf Rangnick opted to leave Cristiano Ronaldo on the bench for a league match. The 37-year-old played the full 120 minutes in the FA Cup defeat against Middlesbrough but he still won't have been best pleased. Nonetheless, a front five of Paul Pogba, Bruno Fernandes, Jadon Sancho, Edinson Cavani and Marcus Rashford didn't look too bad!
Cavani was one of two changes from the Boro match, with David de Gea replacing Dean Henderson between the sticks.
Burnley made a relatively bright start but it was Nick Pope who was called into action first, diving down at his near post to beat away Rashford's fierce effort from a tight angle.
Referee Mike Dean and VAR then took centre stage on 12 minutes when Raphael Varane thought he had headed United into the lead after latching onto Fernandes' free-kick. However – after a good few minutes of deliberation – Dean was called to the monitor and the goal was chalked off for offside.
The Red Devils weren't to be denied a second time, though, and it was a lethal bit of football from Rangnick's men.
Rashford had drifted over to the left and he released Luke Shaw on the overlap. The full-back easily skipped away from Connor Roberts and his low cut-back was leathered home by an unmarked Pogba from 12-yards.
Only a few minutes later, Rashford, now on the right, drilled Shaw's initial delivery back across the box and Josh Brownhill inadvertently prodded into his own net to seemingly provide United's second – only for the linesman to then raise his flag for a Pogba foul on Erik Pieters in the build-up.
Quite how the visitors were only a goal ahead was an absolute mystery and they carried on striding forward with purpose.
On 34 minutes, Cavani looked destined to finally double the lead when Shaw's cross deflected appetisingly to him at the back-post, but Pope made an outstanding reaction save to keep his header out.
Fernandes then became the latest to test Pope with a weak side-footed effort after outstanding work from Diogo Dalot while Rashford again saw a venomous drive well blocked by the Burnley stopper.
The United attack being so fluid seemed to bamboozle the Clarets. Sancho, Rashford, Pogba and Fernandes all seemed to switch to be frequently switching positions on the field and it was making their attacking play so difficult to read.
By contrast, Burnley were very much predictable in their play and they were fortunate to be heading into the break at just one goal down.
It's almost as if they Rangnick has read this script before. His team annihilated Middlesbrough for 45 minutes but they were made to pay for missed opportunities. Lo and behold, the same lightning bolt struck at Turf Moor.
Dyche's men came out of the traps firing and they had an equaliser within minutes of the restart.
Wout Weghorst did brilliantly to spin Varane and Harry Maguire before playing Rodriguez in with a fantastic reverse pass. The 32-year-old showed great composure to take the ball into a more central position before slotting past de Gea. Game on!
The home fans were buoyant and Weghorst soon forced de Gea into a scrambling save, almost catching the Spaniard out with an instinctive effort from 30-yards.
It was an incredible turnaround from the hosts and Weghorst was suddenly involved in all of the action. The Dutchman slammed a shot into the side netting after latching onto Dwight McNeil's cute little pass through the United defence.
Ronaldo came on for the final 22 minutes – replacing the ineffective Cavani – while Cornet hobbled off for Lennon in almost the same action. One team losing their star-man as the other one brings theirs on!
Indeed, the pendulum did seem to swing back after these changes – with Untied players visibly trying to get everybody to up the tempo. Pieters and Brownhill received bookings for their efforts to stem the flow.
As it was, they had Pope to thank for another outstanding reflex save, this time with his leg, to keep out an innovative heel flick from Varane. From Shaw's resulting corner, Ronaldo then rose highest in the six-yard box but he couldn't keep his header down.
Ashley Barnes made a late cameo on his return from injury and young Anthony Elanga came on with Jesse Lingard for one final throw of the dice from Rangnick. Alas, none of the latecomers made much impact and Burnley hung on to their share of the spoils.
The result doesn't do enough to lift Burnley off the bottom but it will help to boost morale. A goal scored and a point gained off one of the best teams in the division.
Rangnick, by contrast, will wonder how his team managed to bungle the win after such a strong opening half. It is the second time in less than a week that his players have done more than enough to win the game, but leave it without the victory.