The victory means Bolton face a trip to Fulham on Sunday.
Wanderers had chances to go ahead in the first half and Johan Elmander hit the post just before the interval.
Klasnic then found the net in the 66th minute, rifling the ball across Mike Pollitt and into the corner having been played in by Mark Davies.
Both managers rang the changes, with Wigan boss Roberto Martinez making nine adjustments to his team.
Pollitt came in for on-loan goalkeeper Ali Al Habsi, who was ineligible to play against his parent club, while Steven Caldwell, Adrian Lopez, Callum McManaman and Franco Di Santo were among those also promoted to the first XI.
Owen Coyle made six changes of his own as David Wheater replaced injured centre-back Zat Knight in the Wanderers defence and Klasnic was handed a start up front in the absence of the cup-tied Daniel Sturridge.
Gary Cahill took the captain’s armband as regular skipper Kevin Davies dropped to the bench.
With the original tie and two Premier League fixtures between the sides this season all resulting in draws, something had to give tonight and it was the visitors who made the more positive start as Lee Chung-yong swung an effort narrowly wide in the third minute.
Seconds later Pollitt parried a low Martin Petrov drive and Elmander finished from close range, but the effort was disallowed for offside.
Wanderers then had a nervy moment at the other end as Ronnie Stam’s cross bobbled off Wheater and Adam Bogdan needed two attempts to keep hold of it.
Di Santo found himself in some space soon after, but the Argentinian tried to square the ball and nobody was up with him.
Wigan were next to make a mistake at the back as skipper Gary Caldwell intercepted a pass by Lee but played in Elmander, who brought a save out of Pollitt.
Jordi Gomez struck a free-kick high and wide in the 23rd minute before Klasnic squandered the best opportunity of the game so far when through one on one, Pollitt getting down to block a shot the Croatian really should have buried.
The hosts then won another free-kick following a high challenge by Fabrice Muamba on McManaman and this time Gomez struck it on target, with Bogdan there to make the catch.
Back came Bolton and Lee chose to cross when he might have shot before Klasnic completely missed the ball when trying to connect with a Paul Robinson cross.
Wanderers kept up the pressure, with Pollitt doing well to tip Petrov’s fierce strike over and Elmander seeing a free-kick come back off the Wigan upright.
McManaman then had to be withdrawn, Victor Moses the man coming on to replace him.
In a scrappy opening to the second period, Muamba sent a shot into the arms of Pollitt and Gomez scuffed an effort wide from the edge of the box.
Moses then showed his pace by bursting into the area but he fired into the side-netting.
Sam Ricketts, who had been trying to keep up with the young forward, was left on the turf and subsequently had to be carried off on a stretcher, with Marcos Alonso taking his place.
Lee then lost possession to Moses, who once again advanced rapidly before laying the ball to James McArthur, but the Scotland midfielder’s shot was the wrong side of the post.
A nice move saw Muamba played in and Pollitt slid in to deny him, before Elmander had the ball in the net, only to be ruled offside once again.
Moments later, though, Bolton were ahead as Klasnic collected Mark Davies’ pass, swivelled in the box and smashed a shot beyond Pollitt.
Wigan looked to make a swift response and after Mohamed Diame’s chip over Bogdan became the third effort disallowed for offside, Gomez cracked an attempt against the bar.
Martinez’s side continued to press for the equaliser, with Di Santo nodding wide and Moses blazing off-target.
Bolton survived, though, and might have added another goal before the final whistle, Lee and Petrov both hitting efforts wide.