Leeds-born Oliver McBurnie twice gave Swansea the lead as the striker haunted his hometown club.
But goals from Kemar Roofe and Pablo Hernandez, the latter a former Swansea player, illustrated the character that Bielsa has instilled in this Leeds side in just a few months at Elland Road.
The Argentinian might actually have been in the other dugout as he had held talks to become Swansea manager in 2015.
McBurnie had only been at Swansea a few months back then after starting out at the Leeds academy.
He subsequently moved on to Bradford and his eye for goal threatened to give Bielsa the sort of rude awakening that an away-day in the Championship can become.
Leeds were forced into a last-minute change when captain Liam Cooper was injured in the warm-up.
Luke Ayling took both the armband and Cooper’s centre-back position,as 18-year-old Jamie Shackleton made his first league start for his hometown club at right-back.
McBurnie’s muscular presence gave Swansea a focal point and Leroy Fer, making his first start since February after rupturing his Achilles, drove them forward from midfield in an enterprising start.
Swansea’s reward came after 24 minutes when Fer, McBurnie and Barrie McKay caught Leeds undermanned on the break.
McKay found Bersant Celina and his pass into the path of McBurnie allowed the Scotland international to beat Bailey Peacock-Farrell, although the Leeds goalkeeper might have done better than to simply get a hand on the ball.
Celina almost doubled Swansea’s advantage and Leeds’s frustration was reflected by first-half bookings for Kalvin Phillips, Roofe and Lewis Baker.
Phillips seemingly paid the price for losing possession in the build-up to Swansea’s goal, withdrawn by Bielsa three minutes after Leeds had fallen behind for the first time this season. Gaetano Berardi then missed the target with a free header, but the equaliser Leeds scarcely deserved arrived five minutes before the break.
Swansea were caught napping by a quickly-taken free-kick and Shackleton beat Martin Olsson to deliver a cross that Roofe tapped home from a few yards out.
Swansea were back ahead six minutes after the restart, McBurnie straining his neck muscles to dispatch Olsson’s cross in magnificent fashion.
Leeds finished strongly and substitute Patrick Bamford broke down the left 11 minutes from time and Hernandez rolled his shot home to earn a draw.
Swansea City: Mulder, Naughton, van der Hoorn, Rodon, Olsson (Asoro 83), Fulton,Fer (Carroll 74),Roberts,Celina,McKay (Montero 64), McBurnie. Unused substitutes: Benda, James, Grimes, Dhanda.
Leeds United: Peacock-Farrell, Shackleton, Berardi, Ayling, Douglas, Phillips (Baker 28), Hernandez, Samuel, Klich, Alioski (Harrison 46), Roofe (Bamford 64). Unused substitutes: Pearce, Blackman, Shaughnessy, Rey.
Referee: Andy Davies (Hampshire).