Yves Bissouma and Florin Andone wrapped up the Seagulls' comfortable Dean Court victory, with both sides making sweeping changes from midweek Premier League exertions.
Knockaert's first goal in 12 games set Brighton en route to another fourth-round tie, with Bissouma bagging his maiden Seagulls strike and Andone his third of the campaign.
Marc Pugh's classy strike briefly had Bournemouth thinking of a second-half comeback, with the midfielder notching his first Cherries goal since August 2017.
But when Andone nodded in after a messy Brighton corner, the game was up.
The FA Cup's place in the pecking order was laid bare by the teams showing a combined 17 changes from their midweek Premier League exploits.
Liverpool loan signing Nathaniel Clyne jumped straight in for an immediate Bournemouth start and showed up well.
Lys Mousset and Jordon Ibe were handed chances to impress up front for the Cherries, with England striker Dominic Solanke waiting in the wings for bigger challenges ahead having signed from Liverpool on Friday.
Neither Mousset nor Ibe have been able to justify their squad numbers of nine and 10 in their time at Dean Court so far – and this outing proved no exception to those struggles.
This tie turned entirely on Mousset's abysmal miss, in a sequence where Ibe's dalliance cost him a clear, point-blank sight on goal.
The French forward fired over from eight yards out with an empty net at his mercy, after Ibe's shot had been blocked by Duffy.
Bournemouth were still stunned by the scale of that sitter when Brighton took the lead moments later.
Steve Cook's weak clearing header dropped to Jurgen Locadia in the box, who checked and found Knockaert.
The French winger blasted home first time, the ball rocketing between Cook's legs to leave Artur Boruc with little chance of making a save.
Three minutes later Brighton doubled their lead, with Bissouma lumping home an ugly, bouncing shot, that Boruc really should have saved.
Instead the ball skipped off the turf eight yards out before nestling in the net via the post.
Locadia could have killed the tie entirely just before the break, but his smart header from Gaetan Bong's cross rebounded off Boruc's left post.
Bournemouth simply had to up the ante after the break, and Pugh finally made good on that requirement with a fine goal.
Andrew Surman fed into the midfielder, who set himself with one touch then stroked home from 20 yards.
Brighton quickly severed any hopes of a Bournemouth comeback however, when Andone nodded in after a corner.