Villa prevailed 2-0 thanks to goals from Fabian Delph and Scott Sinclair, while both sides ended with 10 men as Claudio Yacob and Jack Grealish were dismissed.
Sinclair's well-taken effort five minutes from time prompted a handful of supporters to spill onto the field to join in the celebrations.
With a minute to go in stoppage time, yet more fans returned to the field before being ushered off, and the final whistle from referee Anthony Taylor sparked a mass invasion with players from both sides running to the tunnel to avoid the crowds.
The scenes somewhat soured the day for Villa, who hauled themselves out of the Premier League relegation zone with a 2-1 home win over West Brom in midweek and could be punished by the FA following the actions of some of their fans.
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