Nick Ross got Inverness off to the perfect start with a goal after three minutes that owed a touch to good fortune, before Aaron Doran wrapped up a first home win for manager John Hughes late on.
After the pitch passed two inspections due to the rain that had been falling throughout the day in the Highlands, referee Alan Muir declared the surface playable.
Following Saturday's goalless draw with Hearts, Hughes made two changes to the Caley Thistle starting line-up.
Captain Richie Foran, making his first start since sustaining a shoulder injury in December, came in along with Ross as Doran and Marley Watkins dropped down to the substitutes' bench.
Stranraer boss Stephen Aitken named the same team that had drawn 1-1 at home to Arbroath on Saturday as the Scottish League One side looked to cause a cup upset.
The visitors' chances, however, were dealt an early blow as Caley Thistle took the lead just three minutes in.
Scott Robertson, in trying to cut out David Raven's through pass to Ross, only succeeded in striking the Inverness midfielder with the ball and it rebounded beyond the onrushing goalkeeper David Mitchell.
This helped to settle any pre-game nerves the hosts had and they might have been eased further when Greg Tansey let fly from 20 yards, but Mitchell got down well to his left to make the save.
Stranraer's sluggish start could be put down to their five-hour journey to get to Inverness, although they stirred themselves into action when Jamie Longworth set up Sean Winter to run into the box but Dean Brill made a comfortable stop at his near post.
Inverness, however, were enjoying the bulk of the possession and Graeme Shinnie was not too far away with a rising drive from 18 yards that flew over the crossbar by a yard after the Stranraer defence had failed to properly clear a ball by Ross into the box.
After the part-timers had threatened an equaliser from Andrew Stirling's free-kick that Foran had to stoop low to head away from in front of goal, Mitchell produced a fine reaction save to foil Billy McKay when he was set up 10 yards out by the impressive Ross.
The heavily sanded pitch was standing up robustly to the conditions as the hosts went in at the interval with a one-goal lead.
The opening exchanges of the second half produced little in the way of goalmouth action for either side, despite plenty of Inverness pressure.
McKay's crisp finish was ruled out by the offside flag and Shinnie's powerful run down the left saw his cross partially punched clear by Mitchell to Tansey, but his goalbound shot was blocked away to safety by the Stranraer defence.
As the game approached the latter stages, Stranraer almost levelled matters, Steven Bell trying a speculative effort from 35 yards that Brill had to turn away for a corner.
The Blues' resistance, however, was finally ended in the 90th minute, substitute Doran collecting the ball 20 yards from goal before finding the corner with a low shot past the diving Mitchell.