David Ajiboye, Josh Grant and Luke Armstrong had the Cumbrians 3-0 up inside 16 minutes but the Pilgrims steadied the ship and responded twice within four minutes after half-time through Kieren Donnelly and Lenell John-Lewis.
Connor Teale, Frankie Maguire and Matty Carson all went agonisingly close to an equaliser - but Carlisle’s midweek signing Chris Conn-Clarke scored a brace to seal the home side’s progress.
Graham Coughlan had made seven changes following Wednesday's National League Cup encounter against Newcastle United - and three changes to the side which defeated Gateshead in the National League last weekend.
Matty Carson forced Carlisle keeper Harry Lewis into a smart save inside 30 seconds - but the Cumbrians were soon in full control of this fourth qualifying round clash.
AJIBOYE cut infield to arrow a low left-footer past a stranded Rhys Lovett in the sixth minute - and GRANT touched home a corner delivery nine minutes later.
A minute further on, ARMSTRONG hooked home Josh Williams' near post cross as the shell-shocked Pilgrims battled to save face.
The Pilgrims stemmed the tide - and took advantage after the break. Half-time sub DONNELLY, who replaced Liam Waldock, headed home Marcel Lavinier's cross from the right to give the visitors hope.
The alarm bells were well and truly ringing in the Carlisle rearguard when JOHN-LEWIS powerfully headed home Carson's corner three minutes later.
Lewis subsequently made excellent saves to deny Teale and Maguire and then went walkabouts, only for Morgan Feeney to dig him out of a hole to block Carson's goalbound drive.
Carlisle typically took full advantage and CONN-CLARKE rolled home their fourth goal in their next attack on the hour mark.
The Pilgrims continued to battle manfully against their expensively-assembled hosts - but fell further adrift when CONN-CLARKE added his second 12 minutes from time.
Substitute Georgie Kelly rattled the crossbar late on - but the Cumbrians already had their name attached to ball number 52 in Monday's first round draw.
Carlisle (4-2-3-1): Harry Lewis; Josh Williams, Morgan Feeney, Aaron Hayden, Sam Murray; Josh Grant, Jack Ellis (Sam Hetherington 83); David Ajiboye (Archie Davies 83), Callum Whelan (Jordan Jones 72), Chris Conn-Clarke (Alex Gilliead 83); Luke Armstrong (Georgie Kelly 72). Subs (not used): Regan Linney, Gabe Breeze (gk).
Boston (4-3-2-1): Rhys Lovett; Marcel Lavinier (Ben Grist 86), Connor Teale, Zak Mills, Alex Lankshear (Michael Kelly 86); Dylan Hill, Greg Sloggett (Rocco Coyle 76), Matty Carson; Frankie Maguire, Liam Waldock (Kieren Donnelly 46); Lenell John-Lewis (Jordy Hiwula 76). Subs (not used): Owen Hampson, Bobby Jones (gk).
Referee: Dale Baines.
Attendance: 3,453 (164 visitors).
PIC: Chris Bray.