Podence's impressive strike had sent Wolves into a 12th-minute lead at Villa Park, but Ings came off the bench to register the leveller for Villa in the 78th minute.
The result has left Wolves inside the relegation zone in the Premier League table, with Julen Lopetegui's side now on 14 points from 18 matches, one point behind 17th-placed West Ham United.
Meanwhile, Villa sit 11th, with Unai Emery's side boasting 22 points from their opening 18 games, which has left them just three points behind 10th-placed Chelsea.
Villa had the first half-chance of the contest in the third minute when the ball broke for Leon Bailey inside the Wolves box, but the attacker dragged his effort just wide of the goal.
Wolves then came close down the other end, with Nathan Collins heading a deep cross towards goal, but Ezri Konsa was on hand to clear off the line with Emiliano Martinez beaten.
The visitors did make the breakthrough in the 12th minute, though, when Podence exchanged passes with Joao Moutinho before impressively evading the challenge of Douglas Luiz and then picking out the far corner of the net.
Matheus Nunes had an opportunity to double Wolves' advantage in the 22nd minute when he broke into the Villa box, but Martinez was on hand to make a smart save.
Aside from Podence's opener, both sides found it difficult to show their quality in the final third of the field in the opening half, with just the one goal separating the two sides at the interval.
Villa came close to levelling in the 75th minute when Ludwig Augustinsson volleyed one towards the bottom corner, but Max Kilman was on hand to make a brilliant headed clearance.
The home side did manage to make it 1-1 in the 79th minute, though, when Ings gathered a long pass from Tyrone Mings before finding the back of the net with a smart finish.
Villa had a golden chance to win it in second-half stoppage time when Bailey rounded Jose Sa inside the penalty box, but the attacker could not find the back of the net from a tight angle, with the points ultimately being shared at Villa Park.