Rodriguez's header pegged back Villa back on 68 minutes before Chris Wood quickly cancelled out what the home side hoped would be a winning goal from John McGinn.
The forward got off the mark in his second spell with the Clarets in a Carabao Cup defeat against Sunderland last month but it has taken him seven substitute appearances to open his league account.
All of his previous league goals for Burnley - 31 in total - had come in the Championship.
Rodriguez derived additional pleasure from silencing Villa Park having spent two seasons with their rivals West Bromwich Albion.
"I really enjoyed it [scoring], I love the atmosphere and it's all good fun," he said, as quoted by LancsLive.
"It was an exciting game for the neutrals, end to end but I think fair enough we got a point.
Burnley's vital asset
Sean Dyche's men have only been beaten by big six opposition in the form of Arsenal and Liverpool this season.
That is despite trailing away from home against Brighton (before they levelled the scores in stoppage time) and Villa.
Rodriguez has noted the impressive level of perseverance within the ranks and feels it will pay dividends over the course of the campaign.
"That is the character the lads show and one thing I have definitely noticed since coming here," he said.
“There is a willingness to win and to grind it out and stick together.
“It is really important to the team and you can see that."
A start against Everton?
Rodriguez has featured in all of Burnley's matches but is yet to be selected in the starting line-up.
He has now staked his claim to a greater role with a key goalscoring contribution.
Dyche has proven reluctant to drop Wood or Ashley Barnes to the bench to make room for Rodriguez but may have now found a way to accommodate all three.
He deployed his team in a 4-3-3 shape in the second-half against Villa, with Rodriguez and Barnes flanking Wood.
It remains to be seen, though, whether he will rigidly adhere to a 4-4-2 from the off and only shake things up if the game is going against his side.
Burnley next face 15th-placed Everton at 15.00 on Saturday.