The West Midlands outfit travelled to the North-West having not won at this ground since 2010, and the visiting supporters may have feared the worst when Jacob Bruun Larsen ruthlessly put Burnley ahead.
However, Rayan Ait-Nouri equalised just before half time, before a competitive second half ended with both sides being content with a point apiece.
Burnley have now gone four games unbeaten but remain in the bottom three, while Wolves will be satisfied with not losing ground on the likes of West Ham United and Newcastle United in the race for a European spot, but neither side's cause is helped with a draw.
Although Wolves were able to welcome back Matheus Cunha, the Brazilian started on the substitutes' bench, with Leon Chiwome retaining his spot at the head of the attack.
However, it was Burnley - buoyed from their 2-2 draw at Chelsea - who had the better of the early exchanges, albeit without necessarily testing Jose Sa.
Midway through the first half, a small section of the home fans were forced to leave their seats, a consequence of a loose panel above them, and they were advised to watch the remainder of the match in the fan zone.
As the first half progressed, Wolves had more of the game, Joao Gomes heading wide of the target after a clever Matt Doherty headed-pass back into the middle, but it was Burnley who would take the lead in the 37th minute.
A cross from Dara O'Shea found the run of Bruun Larsen at the back post, and the winger kept his composure to expertly fire a volley into the far bottom corner of the net from 12 yards.
Wolves were huffing and puffing in the final third, with Chiwome struggling to make an impact, but they were able to find an equaliser just before half time from a set piece.
An inswinging free kick from Pablo Sarabia found the run of Rayan Ait-Nouri, who directed a header into the bottom corner of the net from 10 yards out.
Soon after the restart, Ait-Nouri should have put Wolves in front. The Algeria international was released through on goal from a pass from Max Kilman, but Burnley stopper Arijanet Muric made a save that he should not have been allowed to.
Burnley had been second best either side of half time, yet the home team wasted two opportunities to retake the lead, Sa making important saves to deny Larsen and Vitinho.
Cunha was introduced with 13 minutes remaining with the task of finding a winner for Wolves, but it was Mario Lemina who wasted his side's best chance, sending a header wide from an Ait-Nouri cross.
The home side had a late goal from Jay Rodriguez ruled out for offside before added-on time failed to deliver a winner, to the frustration of both sets of supporters.