The Saints, who finished third, will travel to runners-up Harrogate Town on Sunday with the winner clinching promotion to the National League.
Brackley had to do things the hard way against a resolute Bradford Park Avenue side, who had upset Kidderminster Harriers in the play-off eliminator.
But it was star striker Aaron Williams who stepped up when it mattered, poking home in extra time to send the Saints’ faithful into raptures.
The FA Trophy finalists now face another final before their Wembley trip and Wilkin reckons his players have shown they have the mental fortitude to cope with the big occasions.
Following Sunday’s win, he said: “We started the game well, had some chances but didn’t take them and then Bradford tested Danny (Lewis, goalkeeper) from distance because we didn’t keep the ball well enough.
“The boys stayed strong throughout and they had to be. It’s disappointing that there has to be a loser - Bradford have had a great season and they were tough to beat.
“Harrogate have done very well this year and so have we so it should make for a cracking final next weekend.
“Our lads have had a really strong season, mentally and physically, but the boys have proven they’re capable of handling it. We’ve got two big games left - let’s hope we do ourselves justice.”
For more news, reaction and action from across the National League, like us on Facebook and follow us on Twitter and Instagram.