Conor Sammon put the SPL side ahead by lifting a cool finish over goalkeeper Ryan Esson midway through the first half.
He then set up Kelly to net from the edge of the area seven minutes later, with the scorer doubling his tally with a diagonal finish just before the hour.
Calderwood made two enforced changes for the match, with Chris Maguire, the scorer in the Rugby Park side's midweek victory over Celtic, cup-tied and team mate Mehdi Taouil injured.
The duo were respectively replaced by Scott Severin, making his first start, and Sammon.
Inverness manager Terry Butcher named the same starting line-up which defeated Partick Thistle 2-1 in the First Division last week, extending the club's unbeaten run to seven matches.
These sides met at the same stage of the competition last year, with the Highlanders winning 2-0 at home to progress to the quarter-finals.
Adam Rooney of Inverness fell foul of an early offside flag as he shot from Dani Sanchez's cross, while Jamie Hammill had a shot smothered by Esson after good build-up involving Sammon and Kevin Kyle.
Inverness' Richie Foran then picked up possession in midfield, allowing David Proctor to unleash an effort which Bell leapt to claim at the right-hand post.
Severin came close for the hosts on 18 minutes following good work by Hamill on the left-hand side.
Hamill played the ball back from near the touchline out to the midfielder, and his shot from the edge of the area was deflected onto the roof of the net.
Proctor then headed off the line at the near post when Manuel Pascali sent in a shot from Craig Bryson's corner, while Sammon pulled a diagonal effort wide.
As the game entered its second quarter, Sanchez intercepted possession well inside opposition territory and tried a drive which went over the target.
Just a minute later, Kilmarnock took the lead courtesy of a delicate finish from Sammon - Kyle flicked a pass on to his strike partner, who lofted the ball from the right side of the box, over Esson and into the far side of the net.
The same sequence of events played out 10 minutes from the break, when Inverness came agonisingly close to a goal, but immediately fell 2-0 behind.
Foran's shot from the right was parried by Bell and somehow beaten away and scrambled clear just inches from the line with Sanchez attempting to apply the finishing touch.
Instead, the SPL side headed for the other end, Sammon forging a path down the right wing and cutting the ball back for Kelly to score from the edge of the area.
Inverness made a determined start to the second half, Foran trying his luck with a speculative cross which rolled across the face of goal as the visitors tried to apply pressure and reduce the deficit.
Rooney collected the ball near Bll's penalty area and ran in on goal but pulled a diagonal shot past the left-hand post without troubling the goalkeeper.
However, Kilmarnock remained in control.
Kelly fed the ball down the right wing to Tim Clancy near the byline just before the hour mark, but Esson easily palmed away his shot at the near post.
Kelly was more successful on 58 minutes when he doubled his tally to make it 3-0, taking advantage when Lionel Djebi-Zadi failed to deal with Garry Hay's cross to send a diagonal shot from the right past the outstretched Esson and into the opposite corner.
Foran had a shot from the left corner of the six-yard box which Bell blocked, and then forced an impressive save from the Kilmarnock goalkeeper with a 12-yard drive which Bell dived instnctively to beat away round the right-hand post.
Bell made another good save from visiting substitute Eric Odihambo as the game entered its final quarter.
Kelly had a further half-chance with a shot which Esson clutched comfortably to his body.
Inverness sought a late consolation, with Proctor battling past Craig Bryson only for Foran to shoot wide while Munro headed tamely into Bell's arms, and for Kilmarnock, Hamill fired over with a half-chance in the closing stages.