Shaw has been named on the wing while Barrow will partner Hefin O'Hare at centre, with Colin Gregor moving from centre to stand-off.
Among the forwards, Scotland trio Alastair Kellock, Euan Murray and Johnnie Beattie are missing from the squad which claimed a 6-6 draw with Saracens last week.
Jon Petrie replaces Beattie at number eight, while Canada international Kevin Tkachuk moves into the front row with Dan Turner taking over from Kellock at lock.
Scrum-half Graeme Beveridge takes over the captaincy of the team in the absence of Kellock.
Flanker Donnie Macfadyen is set to make a welcome return from injury after being named on the bench for the visit of Cardiff.
Glasgow Warriors team:
15 Rory Lamont
14 Colin Shaw
13 Hefin O'Hare
12 Scott Barrow
11 Thom Evans
10 Colin Gregor
9 Graeme Beveridge (CAPTAIN)
1 Kevin Tkachuk
2 Fergus Thomson
3 Justin Va'a
4 Andy Newman
5 Dan Turner
6 Steve Swindall
7 John Barclay
8 Jon Petrie
Substitutes:
16 Donnie Macfadyen
17 Andrew Wilson
18 James Eddie
19 Moray Low
20 Eric Milligan
21 Sam Pinder
22 Sean Marsden
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