Conor Coady's third own goal of the season put them ahead inside two minutes before teenager Dwight McNeil fired in his third Burnley goal to settle matters in the 77th minute.
It was Burnley's first top-flight win over Wolves since 1963 – coming at the 10th attempt – but more importantly it ended a run of four straight defeats which had left Dyche's side hovering just above the Premier League drop zone.
With crucial games against Bournemouth and Cardiff next up before they face Chelsea and Manchester City, there was a nervous feel around Turf Moor as Dyche was congratulated on his landmark before kick-off, but they were soon settled.
Less than a minute in, Romain Saiss fouled Chris Wood to give Burnley a free-kick on the left.
McNeil whipped the ball in and Chris Wood peeled away from his marker, rounded Rui Patricio and poked his shot on to the post.
Coady, guilty of letting Wood go, compounded the error as he slid in and inadvertently bundled the ball over the line.
Wolves, who left top scorer Raul Jimenez on the bench after his late arrival back from international duty, were slow to muster a response in a game of few clear chances.
Burnley went close again in the 20th minute from another set-piece. Ashley Barnes wanted a penalty after tumbling under the attentions of Willy Boly, but the ball rolled out for Phil Bardsley to strike a powerful shot which was always bending wide.
Wolves were appealing for a penalty of their own in the 22nd minute when Jonny's strike hit Barnes on the arm, but referee Chris Kavanagh waved them away, suggesting he felt Barnes' arm was in a natural position.
Ruben Neves' ambitious volley sliced wide while Ivan Cavaleiro put too much on to a through-ball looking for Diogo Jota, with Wolves looking disjointed in attack.
They began the second half looking much improved as Jota's inswinging cross from the left whistled just wide of Tom Heaton's far post.
Leander Dendoncker then got the better of Ben Mee down the right and cut inside to pull the ball back for Cavaleiro, only to see the Portuguese blaze his shot over the bar.
Adama Traore's cross fell for Jota to strike at goal but James Tarkowski made a vital block.
Wolves were playing their best football of the match, but just barely 10 minutes after the break Nuno Espirito Santo opted for a double change as Jimenez and Matt Doherty replaced Dendoncker and Cavaleiro.
If anything the changes robbed Wolves of momentum, but they were still on top with Burnley pegged back in their own half and the Turf Moor crowd sitting nervously in their seats.
Burnley had to wait for a set-piece to get a sight on goal and it was a good one when it came, with Boly needing to stretch to block Mee's header from Ashley Westwood's free-kick.
The first effort on target for either side finally arrived in the 74th minute, but Jonny's header from Joao Moutinho's dinked cross was easy enough for Heaton to gather.
Moments later, McNeil settled it. Picking the ball up just inside the Wolves half, the 19-year-old drove at goal and, as defenders stood off him, struck a left-footed shot across goal to beat Patricio and seal three important points.