The game at Tolka Park provided a second clash between the champions of the Irish Premiership and League of Ireland Premier Division in as many seasons after Shamrock Rovers thumped Larne 4-1 in last year's Uefa Conference League.
And while the Republic of Ireland side again asserted their dominance in the first half of this one, Linfield held firm through the opening 45 minutes despite a few scares.
Caffrey looked lively throughout for Shelbourne and came closest to opening the scoring in the early exchanges when he got on to the end of a near-post corner only to see his flicked header come back off the crossbar.
The same player then drew a brilliant save from Linfield keeper Chris Johns from point-blank range after defender Euan East had slipped in the build-up.
As Shelbourne piled on the pressure, they thought they would have the opportunity to take the lead from the penalty spot when Ben Hall was adjudged to have handled Jonathan Lunney's effort from the edge of the box, but the decision was overturned by VAR.
While admirably organised without the ball, in the absence of playmaker Joel Cooper who joined Coleraine over the summer, Linfield offered little attacking threat, although they did have a few chances of their own in the goalless first half.
First Ethan McGee drove an effort wide after latching onto a loose pass across the box before, in the final minutes before the break, Callumn Morrison's looping effort failed to catch out Conor Kearns in the Shelbourne goal.