The away side, on the other hand, make the trip in search of back to back victories in competitive matches for the first time since February.
Match preview
St Johnstone just about did enough to edge out Brechin in a 2-1 triumph as the teams locked horns at Glebe Park on Saturday.
Craig Levein's men went into the dressing room with a narrow advantage thanks to Graham Carey's late first-half penalty before Kyle Cameron doubled the lead which was eventually halved by Spencer Moreland.
As a result, The Saints have moved up to second place in Group F, one point shy of East Fife who have, however, played one game more.
Awaiting them next is the challenge of an opposing outfit who are without a victory in the last 15 meetings between the teams since winning the first encounter back in 2006.
Still winless in nine home matches since the turn of the year, Saturday's hosts have laboured in front of their fans and will be keen to finally end their barren run.
Greenock Morton recovered from an opening defeat at the hands of East Fife to claim a 1-0 win against Brechin in Tuesday's matchup at Cappielow Park.
Cammy Ballantyne broke the deadlock just shy of the hour mark and that was enough to get the job done for Dougie Imrie's men in a tight affair.
With three points amassed from two matches, The Ton sit third in Group F, level on points with their next opponents ahead of the meeting in Perth.
Having managed just one win from their last eight away outings leading up to this weekend, Saturday's visitors have been largely underwhelming on the road and now set out in need of a pick-me-up.
St Johnstone Scottish League Cup form:
W
St Johnstone form (all competitions):
L
L
L
D
W
W
Greenock Morton Scottish League Cup form:
L
W
Greenock Morton form (all competitions):
L
W
D
L
L
W
Team News
Kyle Cameron had a debut to remember for St Johnstone last time out as he netted the all-important winning goal to lead his new employers to victory.
New arrival Uche Ikpeazu is still nursing a muscle injury and the forward is unavailable for selection this weekend.
Greenock Morton's Morgan Boyes was hauled off with an injury scare in the first half of the win over Brechin, making the Welshman a major doubt for this one.
Should Boyes be deemed unfit to participate, 39-year-old Kirk Broadfoot will be the favourite to come into the visitors' lineup.
St Johnstone possible starting lineup:
Rae; Cameron, McClelland, Sanders; Carey, MacPherson, Essel, Wright; Smith, Kimpioka, McPake
Greenock Morton possible starting lineup:
Mullen; Ballantyne, Baird, Boyes, Delaney; Davies, Lyall; Moffat, Blues, Garrity; Reynolds
We say: St Johnstone 3-1 Greenock Morton
Given St Johnstone's dominance of this fixture, the home side are huge favourites to secure maximum points and we fancy them to live up to expectations by cruising to a routine win in the end.