The Baggies striker netted in stoppage time to claim a 2-2 draw for the hosts at the Hawthorns.
Rodriguez appeared to handle the ball and Smith said he knew it should have been ruled because ALL off his 11 players saw him use his hand.
“We’ve come toe-to-toe with a team that got relegated last year, on their home patch,” Smith said.
“We probably should have put the game to bed with the chances that we’ve had.
“But they’ve scored a goal that should have been disallowed.
“It’s something, when you get a reaction from all 11 players. You know something’s amiss.”
Villa went ahead through Anwar El Ghazi’s deflected effort after 12 minutes and Dwight Gayle slotted home to level 16 minutes later.
El Ghazi put Smith’s back in front with a stunning effort on the hour mark and Villa had numerous chances to put the game to bed.
However, they were punished for their wastefulness when Rodriguez nipped in to bundle home Matt Phillips’ right-wing cross.
Villa were heading for the play-off places but remain eighth in the Championship and are two points off the top six.
“We worked so hard for this game,” Villa’s star man El Ghazi 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.”
West Brom stay third and are well in the hunt for the automatic promotion spots.
On Rodriguez’ goal, Baggies boss Darren Moore said: “It’s all happened so quick and we’ve ended up getting a point from it.
“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.”