Sunderland had Enzo Le Fee making his debut after joining on loan from Roma, and the midfielder showed a glimpse of what he can offer with some terrific skill and a fabulous pass to put Isidor away for their best chance, but he struck the outside of the post.
If Isidor should have put that one away, it was nothing compared with what was to follow.
He grabbed the ball after he had been brought down by CJ Egan-Riley, possibly outside the 18-yard box on first glance, and went for power, but Trafford guessed right and parried it away.
The two were then protagonists again deep into injury time as Anthony Backhouse awarded a second penalty kick after Dennis Cirkin was brought down by substitute Oliver Sonne.
Isidor grabbed the ball again but was made to wait as Trafford needed treatment and was then booked for timewasting before going went for the same spot, albeit closer to the corner, but Trafford repeated his heroics by again getting down to it to make a brilliant save.
These were the first spot-kicks saved by Trafford since the 2023 European Under-21 Championship final when he saved Abel Ruiz's effort in injury-time as England defeated Spain.
As Trafford basked in adulation, Isidor, Sunderland's top scorer this season, played the final minutes in a daze, unable to comprehend what had happened and he needed consoling at the final whistle.
What impact this has on his and his team's promotion hopes, only time will tell.
Burnley head coach Scott Parker said:
"It's a massive point for us in the scheme of where the game was. Two penalties and the game could have looked very different if not for James Trafford and the saves he's made.
"Not many words can describe what he's produced. I don't think I've been involved in a game like that one at the end.
"Both were very good penalties and Traff has flown across his goal. Everything about him in terms of those big saves, we owe him one tonight."
Sunderland head coach Regis Le Bris said:
"We created five or six chances so we can be disappointed that we didn't score, but it shows that we can play a tough team away.
"Wilson is disappointed, like the team. But football is like that. You can make mistakes but you have to learn from them. It's the best way to improve and grow. I am sure he can put this behind him, 100%.
"You have to deal with difficult experiences. The goalkeeper made two good saves, they weren't poor penalties."