The Clarets had opened the scoring just before the half hour mark through New Zealand international Cameron Howieson before Jason Gilchrist doubled Burnley's lead with a well taken header shortly after the restart.
Bolton did throw men forward in the final third of the game in a bid to rescue a result, but Jamie Frost's late header extinguished any hopes of a comeback to ensure that the hosts ran out comfortable winners.
It was clear to see from the outset that both teams were up for the fight, with Steven Hewitt going close for Burnley while Chris Lester forced Nick Leversedge into a smart save in the seventh minute.
Ryan Noble looked set to hand the Clarets the lead with ten minutes gone after Gilchrist had whipped a dangerous ball across the face of goal, but he narrowly failed to connect with only Jonny Sutherland to beat.
Gilchrist was again in the thick of the action just moments later as he unleashed a powerful effort from an acute angle, but the Bolton keeper was equal to his shot, while Howieson blasted his 30-yard effort narrowly over the crossbar with 25 minutes on the clock.
Having enjoyed the majority of the chances inside the opening third of the game, it came as little surprise when the hosts opened the scoring in the 27th minute through Howieson who slammed the ball past Sutherland on the half-volley.
Their lead was almost doubled just three minutes later as Hewitt found himself in acres of space, but he was unable to control the ball with the goal at his mercy.
After withstanding a tirade of Burnley pressure, the Whites almost snatched a way back into the clash through Luke Woodland’s ambitious effort, but the ball flew inches over Liversedge’s crossbar.
As the half time whistle drew near, Noble looked to have extended the Clarets’ lead as he powered the ball towards goal, but Glenn Matthews was in the right place at the right time to deflect his effort away for a corner.
After ruling proceedings during the first period, Burnley powered forward in a bid to extend their lead as the second half began and it wasn't long before their efforts were rewarded as Gilchrist headed home Cameron Dummigan's cross in the 56th minute.
Having fallen further behind, Wanderers made changes in a bid to rescue the clash with Georg Iliev being introduced into the fold on the hour mark in a bid to hand the Whites some extra firepower.
Gilchrist almost turned provider in the 72nd minute as he played the ball into the feet of Noble, but the striker proceeded to fire his effort over the crossbar with it seeming as though a third goal was imminent.
As the game wore on chances became few and far between for both sides, but the Clarets weren't quite finished as Frost's lofty header eluded Sutherland the put the result beyond doubt with eight minutes left to ensure that the home side would progress to the last eight of the competition in style.
Burnley: Liversedge, Dummigan, Coleman, Anderson, Conlan, Frost (Jackson, 87), Howieson, Hewitt, Love (Lowe, 80), Gilchrist, Noble
Subs not used: Nizic, Errington, Brownhill
Bolton: Sutherland, White, Matthews (Hendrie, 60), Holding, Bolger, Kelly, Lester, Woodland, Wilkinson, Clough (Iliev, 60), Youngs
Subs not used: Bailey, Walker, Hinchcliffe