The England Under-21 striker netted this fourth goal in five matches after Joe Cole netted for the first time in 23 months late in the first half
Having had the better of the first half, Villa took the lead as Joe Cole poked home Andreas Weimann's through ball to net his first goal for 23 months.
Ings tested Brad Guzan moments later from long-range, and the goalkeeper had to be alert again after the break to deny Michael Kightly from close range.
But the American was eventually beaten as Jores Okore brought down Lukas Jutkiewicz on the edge of the box and Ings converted from 12 yards.
Villa's preparations were hardly helped by Roy Keane's decision to relinquish his role as assistant manager on Friday, though the visitors did look threatening during the early exchanges.
In fact, Lambert's men went agonisingly close to taking the lead in the sixth minute as Okore flicked Ashley Westwood's corner towards goal, only to see Dean Marney clear from underneath the crossbar.
Villa continued to pile the pressure on and caused more problems from a corner a few moments later, with Marney again decisive as he blocked Ciaran Clark's goal-bound header.
Burnley eventually created their first chance of the match in the 14th minute as Kightly cleverly flicked the ball over his own head and then shot right across the face of goal.
Gabriel Agbonlahor really should have put Villa ahead shortly after, though, as the rapid forward linked well with Cole and then shot straight at Tom Heaton when one-on-one.
Villa eventually took a hard-earned lead seven minutes before the break; Cole poking inside the post after latching on to Weimann's well-weighted pass.
Burnley started the second half brightly and almost restored parity in the 56th minute as Guzan denied Kightly from close range after the forward dribbled past a couple of defenders.
Villa remained a very potent threat on the counterattack, however, and Heaton was forced into action two minutes later to palm a Weimann effort around the post.
Burnley were finally rewarded for their late pressure as Okore hauled second-half substitute Jutkiewicz to the ground in the box and Ings confidently smashed home the penalty.
The hosts continued to pile men forward and twice they went close in stoppage time as Ings hit the post and then Jutkiewicz headed just wide, but ultimately they had to settle for a point.