North End survived the sending-off of Shane Cansdell-Sherriff to clinch a 3-3 draw at Bury before winning 5-4 on penalties in a dramatic encounter, with Graham Cummins scoring the decisive spot-kick.
Westley said: "The outstanding feature from us wasn't our tactical play or our technical play. It was our fight and desire and a team ethic of the type I probably haven't seen since I've been at Preston."
He added: "Real fighting spirit got them through and those five penalties said everything about the team. I told them as they stepped up that it wasn't about luck, it was about character. The strongest characters win."
Jack King put Preston ahead from Joel Byrom's corner but Bury levelled after the break through a controversial Joe Skarz header which left Westley fuming.
Westley added: "Andy Procter went down, (Thorsten) Stuckmann threw the ball out and we let David Healy have the ball so he could give it back to the keeper but he turned and attacked us. People talk about sport and respect but then we're allowing situations like that to go on. It's disgusting and punishments should be handed out."
Procter rifled Preston back into the lead after a poor punch by Trevor Carson before Matt Doherty smashed home Bury's second from a tight angle. Cansdell-Sherriff received a second yellow card but Stuart Beavon capitalised on another Carson error to force the ball home in the final minute.
A stunning Steven Schumacher free-kick brought Bury level again but a painful shoot-out defeat left manager Kevin Blackwell questioning his luck.
He said: "Some of the football we played was terrific and I'm just sick we weren't able to finish off a terrific display. I've got no complaints with the players. At the moment they're not getting the reward for their effort or the quality of their play."
Blackwell accused Westley of refusing to shake his hand after the match. He added: "I'm disappointed with Graham Westley because I offered him my hand and he turned it away. I'm a manager of stature in this game compared to Graham and if I offer my hand I expect him to take it. He's won so he should be magnanimous and move on."