Villa started aggressively and thought they had broken the deadlock in the 25th minute with John McGinn slotting home from close range but the goal was chalked off after a VAR review adjudged Conor Hourihane to have been offside in the build-up.
The hosts continued to test Burnley's defensive line and took the lead through a blistering counter-attack, with Dutch forward Anwar El Ghazi smashing home a half volley from a tight angle.
Backing his aggressive instincts, Burnley boss Sean Dyche introduced Rodriguez in the second half and he delivered with a thumping header for the equaliser.
Villa took the lead for the second time in the 79th minute when McGinn volleyed home from close range but Burnley struck back inside two minutes through Wood, who escaped his marker to head home.
Villa failed to move out of the relegation zone after a fourth consecutive league match without a win.