Swansea City suffered last-gasp heartbreak at Turf Moor as Jay Rodriguez's stoppage-time penalty gave Burnley all three points.
The Swans had looked set to back up their wins over Oxford and Watford with a point against a Clarets side fancied for promotion following their relegation from the Premier League.
But they were denied at the last as referee David Webb adjudged that Matt Grimes had handled as he contested a looping ball in the box, and veteran forward Rodriguez found the net from the spot.
It was harsh on the visitors, who had chances of their own to break the deadlock. Liam Cullen saw his header hacked off the line, while Joe Allen and Florian Bianchini were unable to finish good chances.
Lawrence Vigouroux -facing his old club- made a handful of saves, while the visitors defended stoutly as a team with centre-halves Ben Cabango and Harry Darling outstanding once again.
But, despite the agonising finale, Swansea once again showed an ability to go toe to toe with the fancied teams in the division.
The Swans had shown two changes from the midweek win over Watford as Joe Allen and Josh Tymon – the latter back from suspension – came in for Liam Cullen and Kyle Naughton.
The hosts, meanwhile, were unchanged from their goalless draw at West Bromwich Albion on Thursday night.
Williams had spoken of this being one of the toughest assignments in the Championship, but his side opened up brightly, controlling possession and posing questions of the Clarets rearguard.
The hosts responded, captain Josh Brownhill sending a rasping drive over the Swansea bar after patient build-up around the edge of the Swansea area.
Allen then blocked an effort from former Swan Connor Roberts after the defender had been picked out by a Burnley corner.
The Swansea midfielder then spurned a good opportunity after Matt Grimes and Ronald had combined down the left for the latter to roll a ball into Allen’s path as he arrived in the area, but he sent his final effort well wide.
Zian Flemming sent a firm header wide of the target from a corner of Burnley’s own before Brownhill took aim from distance, but Lawrence Vigouroux securely held his firm drive as the Swansea keeper returned to his old stomping ground.
Brownhill was then unable to turn in a Lucas Pires cross as the game closed in on half-time, while Zan Vipotnik blasted into the side-netting after Goncalo Franco had picked out his run.
A piece of brilliance from Josh Key then started a good spell of Swansea pressure, as he chopped inside to leave a defender in his wake and surge into the Burnley half.
Tymon came agonisingly close to picking out Vipotnik with a low cross a few moments later, before Vigouroux parried away Luca Koleosho’s strike as the half ended goalless.
Ronald headed into James Trafford’s gloves from a Key cross early in the second half while, at the other end, Jeremy Sarmiento lashed a strike over the bar from an intelligent Roberts pull-back.
Grimes came close to picking out Ronald with a superb raking pass, but the bounce beat the Brazilian and Trafford was on hand to claim.
Flemming then sent a looping header straight to Vigouroux before Swansea went the closest of either side to breaking the deadlock as substitute Cullen rose to meet Key’s cross.
His header beat Trafford, but Maxime Esteve made a tremendous goalline clearance to prevent the ball finding the net.
Key send a shot straight into the keeper’s midriff as the final 20 minutes approached, and Tymon then dragged wide from a Ronald lay-off.
The next big chance fell for Bianchini when a block ran kindly to put him in on goal, but he dragged his effort wide.
Vigouroux then made a tremendous reflex save as to CJ Egan-Riley's near-post header, but could not deny Rodriguez from the spot at the last.
Swansea City: Lawrence Vigouroux; Josh Key, Ben Cabango, Harry Darling, Josh Tymon; Matt Grimes (captain), Gonçalo Franco, Joe Allen (Liam Cullen 62); Ronald (Cyrus Christie 85), Myles Peart-Harris (Kyle Naughton 77), Žan Vipotnik (Florian Bianchni 62).
Substitutes: Jon McLaughlin, Kristian Pedersen, Nathan Tjoe-A-On, Nelson Abbey, Azeem Abdulai.
Burnley: James Trafford, Maxime Esteve, CJ Egan-Riley, Jeremy Sarmiento (Hannibal 77), Josh Brownhill (captain), Jaidon Anthony, Connor Roberts, Zian Flemming (Jay Rodriguez 77), Lucas Pires (Bashir Humphreys 67) , Josh Cullen (Josh Laurent 77), Luca Koleosho (Enock Agyei 67).
Unchanged Substitutes: Vaclav Hladky, John Egan, Andreas Hountondji, Owen Dodgson.
Referee: David Webb
Attendance: 18,717