The Baggies forward found himself unmarked in the box deep into injury time.
But he appeared to handle the ball twice as it trickled into the net.
The 29-year-old initially seemed to touch the ball as he collided with Orjan Nyland.
As Rodriguez fell he then prodded the ball again with his hand and directed it into the net.
Villa fans were raging after the final whistle went.
One Twitter user said: “Absolutely robbed by the handball at the end but it does show how far we’ve come so quickly under Dean Smith. Can’t be too disappointed, we’ll be ok playing like this.”
Another added: “How has the ref given that and how has the assistant not seen that and he looking straight at it #blind.”
Rodriguez admitted to Sky Sports after the game that he did handle the ball.
Aston Villa ace Anwar El Ghazi was gutted to drop two points in such a controversial fashion.
“We worked so hard for this game,” he said.
“We’re going home like we lost the game.
“We’ve got a feeling that we lost the game now.
“We’re really disappointed but we have to look forward now.”
Despite the controversy, West Brom boss Darren Moore explained he was relieved to rescue a point.
“[On the controversial equaliser] It’s all happened so quick and we’ve ended up getting a point from it,” he said.
“But I’m delighted because of the way the boys stuck at it and kept going right to the end.
“In a hotly-contested game, we played some good stuff.”