Manager Frank Lampard kept just two of his starting lineup from Wednesday's fixture with Bohemians as Willy Caballero and Michy Batshuayi remained in the team, but that did not prevent Chelsea from quickly finding a rhythm.
The Belgium international then struck the top of the crossbar with a thunderous effort, but it did not take much longer for the West Londoners to go in front.
Mateo Kovacic picked out the run of Mount - who had progressed through the centre of the defence - and the youngster took a touch before dinking the ball into the far corner from close range.
St Patrick's were offering very little at the other end of the pitch, and it soon became a struggle for the Irish side to remain in close proximity with a Chelsea side who were determined to impress their new manager.
Tammy Abraham curled over from the edge of the area and although Murphy did superbly to push a Ross Barkley strike onto the post, the veteran goalkeeper could only allow Palmieri's well-struck effort from 20 yards to go into the middle of the net.
However, Murphy went some way to redeeming himself with a stunning stop to keep out Batshuayi's volley from point-blank range, while the 34-year-old was also equal to a 20-yard free kick from Barkley.
Batshuayi was provided with yet another chance just before the break, only to fire over the crossbar, before Abraham somehow failed to find the back of the net, instead firing too close to Murphy after being played through on goal.
Kasey Palmer and Giroud both failed to make the most of half-chances soon after the restart, before Chelsea upped the tempo after the hour mark with Pedro and Kenedy both having low efforts kept out by Murphy.
Nevertheless, there was almost an inevitability about a third goal eventually arriving, and it did when Giroud was on hand to volley home at the back post after a delightful cross from Kenedy.
The pace of the game dropped during an uneventful closing quarter, but Chelsea did add a fourth with three minutes remaining as Giroud found the bottom corner from 12 yards after being allowed to advance into the penalty area.
ST PATRICK'S ATHLETIC XI: Muprhy; Cleary, Webster, Kelly, Bermingham; Forrestor, McCabe, Markey; Clarke, Walker, Doona
CHELSEA FIRST-HALF XI: Caballero; Zappacosta, Tomori, David Luiz, Palmieri; Jorginho, Mount, Kovacic, Barkley; Abraham, Batshuayi
CHELSEA SECOND-HALF XI: Cumming; Azpilicueta, Zouma, Christensen, Alonso; Bakayoko, Gilmour; Pedro, Palmer, Kenedy; Giroud