Danish striker Simon Makienok completed his hat-trick in extra-time as the Lilywhites won 3-2 to reach round four.
Three days earlier, Preston conceded three late goals to lose 5-0 at Brentford in the Championship.
"It showed there was a reaction from the squad after a crazy 15 minutes on Saturday," said Grayson.
He told BBC Radio Lancashire: "I thought we played well, we deserved the victory over the course of the game. We created some good chances but had to withstand some pressure and then with five minutes to go, we've showed the spirit and character to get back into it again.
"A lot of those players have been chomping at the bit to come into the team and show what they are capable of, full credit to them."
Forward Makienok, who is 6ft 7ins tall, moved to Preston on a season-long loan from Italian Serie A side Palermo in June.
The former Brondby man scored five goals on loan at Charlton last season and his goals at Bournemouth were the first since his move to Deepdale.
"It's my first ever hat-trick and scoring a left foot, right foot and a header, it couldn't be any better," he said.
"We were playing a strong side even though they changed a lot of players. You get a lot of confidence from this and now I want to push on and take my chances when they come."
Preston manager Simon Grayson has praised the way his side responded to a heavy defeat in the league to knock Bournemouth out of the EFL Cup.