Paul Pogba’s 18th minute goal put United ahead after 45 minutes, in a half that saw VAR used to rule out Raphael Varane’s header.
Rodriguez equalised for Burnley three minutes following the restart with a composed finish inside the box to earn a point from Wout Weghorst’s neat assist.
Clarets’ goalkeeper Nick Pope made the 200th league appearance of his career, as Ralf Rangnick’s Red Devils visited East Lancashire.
Dyche named two changes to his starting line-up, with Josh Brownhill and Rodriguez came in for Dale Stephens and Aaron Lennon, while Ashley Barnes was named as a substitute.
Before kick-off, both teams observed a minute’s silence which marked our memorial fixture to supporters who have passed away.
First half
Inside the opening nine minutes, Pope was forced to palm Marcus Rashford’s fierce strike around for a corner.
United had the ball in the net on 13 minutes, when Varane headed Bruno Fernandes’ free-kick across goal and beyond Pope, but following a lengthy VAR check, referee Mike Dean disallowed the goal as Harry Maguire was offside.
But the visitors from Old Trafford did take the lead with 18 minutes played. Luke Shaw pulled a ball back to Pogba in the box, and United’s French international struck right-footed high past Pope.
For the second time, United saw a deflected Rashford finish ruled out for offside.
Midway through the half, Pope made two big saves, firstly, denying Edinson Cavani’s close-range header with his knee and Rashford’s powerful shot from his legs.
Second-half
The Clarets were level after 48 minutes through an excellent finish from Rodriguez. The centre-forward beat Maguire to meet Weghorst’s neat pass, to hit a composed right-footed effort in the box past De Gea.
Buoyed by their goal, Burnley went close to going in front, when Weghorst tried his luck with a dipping half-volley from distance, heading for the bottom left-corner, de Gea superbly tipped wide.
From McNeil’s resulting corner, which United half-cleared but only to the feet of Erik Pieters, only for the Dutch defender to miss the target from 20 yards.
On the hour, Weghorst was found by McNeil’s pass, and from a tight angle, thumped the ball into the side netting.
De Gea had to hold onto McNeil’s sighter through a crowd of players, and Connor Roberts’ drive was too high, as Burnley pushed for a second.
Dyche made his first change of the contest with just under 20 minutes left, as Maxwel Cornet hobbled off and was replaced by Aaron Lennon.
Captain Ben Mee then threw himself in the way of Jadon Sancho’s half-chance, prior to Pogba being thwarted by a combination of the skipper and James Tarkowski.
Pope then produced another outstanding save to prevent Varane from scoring United’s second with a header at the back post.
Substitute Cristiano Ronaldo could only guide Fernandes’ corner over Pope’s crossbar when unmarked.
Late on, Weghorst hooked a shot from McNeil’s cross which was caught by de Gea.
Burnley had to deal with late pressure, with Mee blocking Fernandes’ drive and Maguire glancing a corner past the post, to earn a resolute point.
The Clarets are back in action on Sunday, when Liverpool visit Turf Moor (KO 2pm).
Burnley: Pope, Pieters, Mee ©, Tarkowski, Roberts, Westwood, Brownhill, McNeil, Cornet (Lennon 69), Rodriguez (Barnes 89), Weghorst
Subs not used: Hennessey, Lowton, Collins, Long, Bardsley, Cork, Stephens
Manchester United: De Gea, Dalot, Maguire, Varane, Shaw, Pogba, McTominay (Lingard 78), Rashford (Elanga 88), Fernandes, Sancho, Cavani (Ronaldo 68)
Subs not used: Henderson, Lindelof, Jones, Wan-Bissaka, Matic, Mata
Referee: Mike Dean
Attendance: 21,233
Goals: Pogba, 18: Rodriguez, 48