Both Villa and Newcastle have failed to win the competition since the 1950s and another opportunity for Unai Emery's men to end a 30-year wait for silverware vanished after a first-half red card for goalkeeper Marco Bizot.
Tammy Abraham had fired the home side in front with his first goal since returning to Villa last month.
But the game swung in the Magpies' favour when Bizot charged off his line to wipe out Jacob Murphy and deny a clear goalscoring opportunity.
Newcastle's fightback saved the officials from an embarrassing moment, becoming more controversial when Lucas Digne's handball, which was clearly inside the box, was given as a free-kick instead of a penalty.
With VAR not in operation at this stage of the competition, the decision could not be reviewed.
But Newcastle levelled from the resulting free-kick anyway as Tonali's strike deflected past the helpless Emi Martinez.
Tonali has been linked with a move to Arsenal come the end of the season and the Italian showed why he is in demand with a blistering strike from outside the box to turn the tie around.
Nick Woltemade then netted his first goal in 15 games to take Newcastle, who last lifted the trophy in 1955, into the last 16.