The striker, who had been one short of the three-figure milestone for almost a year, since netting in a 3-0 win over Crystal Palace at Selhurst Park, restored parity just minutes into the second half.
Deadline day signing Wout Weghorst was magnificent in the build up, taking two players out of the equation with a neat touch, before slipping a sublime reverse pass into his strike partner, who stepped across his marker before slotting the ball past David De Gea.
The Dutchman almost opened his account for the Clarets moments later when allowing the ball to drop before thumping it goal-wards from distance. However, De Gea got down impeccably to get a strong hand to the attempt as the ball arrowed towards the corner.
Those moments marked a significant shift from the home side's first half display as Ralf Rangnick's Red Devils blew them out of the water.
They had the ball in the net three times, though only one registered. That came from Paul Pogba in the 18th minute as the Frenchman passed the ball into the roof of the net from Luke Shaw's pull back.
Raphael Varane had guided a header past Nick Pope before hand, but the goal was chalked off following VAR's intervention after Harry Maguire was deemed to have obstructed Rodriguez as Bruno Fernandes' set-piece was delivered.
Josh Brownhill later inadvertently poked the ball into his own net from Marcus Rashford's pass, though Pogba had already been penalised for fouling Erik Pieters from Shaw's original delivery.
Here are the ratings.
9. Josh Brownhill 5.5
Inadvertently poked the ball into his own net in the first half, though Pogba had already been penalised for a foul on Pieters. Couldn't make an impact in the first half, distribution was questionable at times after the break, but he worked tirelessly to add a protective layer to his defence. Photo: Nathan Stirk
10. Wout Weghorst 7.5
Gave a much clearer indication as to what he can offer Burnley in this second half of the season. His footwork was a sight to behold at times, the striker's touch and assist for Rodriguez's equaliser was outstanding and he almost opened his account with a clever and confident attempt from distance. Photo: Alex Livesey
11. Jay Rodriguez 7
Continued from where he left off against Watford. Did everything he could to get involved in the first half, despite very little service, but his influence grew after the break. Took the 100th league goal of his career exceptionally well and proved a handful for Maguire. Photo: Clive Brunskill
12. Aaron Lennon 5
Replaced Cornet after the Ivorian limped off with cramp and partnered Roberts on the right, with McNeil shifting over to the left. Helped Roberts tighten things up late on and tried to provide an out-ball for the Clarets when United gambled in search of a winner. Photo: Clive Brunskill