Fernando Llorente sank Burnley with two towering headers before Martin Olsson admitted Spain's World Cup-winning striker has his own Swansea team-mates running scared.
Llorente settled a controversial Liberty Stadium clash as his stoppage-time winner secured a 3-2 victory to lift Swansea five points clear of the Premier League relegation zone.
It was Llorente's 10th league goal since the end of November, and only Tottenham striker Harry Kane and Arsenal forward Alexis Sanchez have scored more in that time.
And Swedish full-back Olsson revealed that no-one messes with the 6ft 4ins former Juventus and Sevilla star on the Swansea training ground.
"Fernando is the same in training, he wins absolutely everything in the air," said Olsson, the January signing from Norwich who scored for the second successive home game in Swansea's comeback win.
"He is a nightmare when we do attack versus defence in training. All the lads try to avoid him.
"Even if you try to elbow him a bit to put him off he is a strong, strong guy.
"He has been at big clubs before, but he is a really humble guy and he works so hard. The whole team's style of play suits him."
Swansea had to overcome a real sense of injustice to continue their remarkable revival under Paul Clement.
Llorente's opener was quickly cancelled out by an Andre Gray penalty, incorrectly awarded by referee Anthony Taylor.
The Cheshire official, who it is understood returned from a stag do in Marbella at the start of the week, felt a Swansea player had handled inside the box, but t elevision replays clearly showed that the ball had in fact struck the arm of Burnley striker Sam Vokes.
"When the penalty decision went against us we did not want to feel sorry for ourselves," Olsson said.
"We did not want the referee to decide the game, so we just kept going and managed to win.
"We have got the momentum to get out of this (trouble). It does not matter how we play, the mentality is there and the gaffer and the staff have it too.
"We feel that we are going to get something out of every game and it is in our own hands now, we are not relying on anyone else."
Having been bottom of the table with 12 points from 19 games, Swansea have won five of Clement's eight matches to give themselves some breathing space from the drop zone.
Burnley are four points better off than Swansea, but the Clarets continue to enjoy little joy on their travels.
While Sean Dyche's side have turned Turf Moor into a fortress, they have picked up only two away points on the road at Manchester United and Hull.
Burnley's next two games are at Liverpool and Sunderland, and defender Ben Mee said: "It's not a case of relying on our home form.
"We try to perform away from home and we did pretty well last week at Hull and in previous games against some big teams.
"We want to win as many games as we can with 11 games to go.
"We went on a good unbeaten run at the end of last year and we need to have that same frame of mind this season."