Hooker Sean Cronin returns from Six Nations duty and replaces Adrian Flavin, while Ray Ofisa starts at number 8 ahead of Mike McComish.
The back-line remains unchanged as Troy Nathan and Niva Ta'auso make up the centre partnership and Fionn Carr, Brian Tuohy and Gavin Duffy complete the back-three.
Johnny O'Connor continues to captain the side from openside flanker, with fellow Galway native Andrew Browne starting at blindside flanker.
"We want to build on the momentum from last week and the good victory we achieved over the Scarlets," said Elwood.
"Obviously there is history between us - we beat Glasgow over there this season and they will come smarting from their defeat last week. Where we both are in the league table suggests this is a potential eight-pointer.
"Our win over the Scarlets was very important. It's okay patting yourself on the back saying, 'another moral victory, another losing bonus point, we are making improvements.'
"Yes, we are, but the game is about winning and that's why beating the Scarlets was so important. Now we want to push on from here."
CONNACHT (v Glasgow Warriors): Gavin Duffy; Brian Tuohy, Niva Ta'auso, Troy Nathan, Fionn Carr; Ian Keatley, Frank Murphy; Brett Wilkinson, Sean Cronin, Jamie Hagan, Michael Swift, Mike McCarthy, Andrew Browne, Johnny O'Connor (capt), Ray Ofisa.
Replacements: Adrian Flavin, Ronan Loughney, Rodney Ah You, Bernie Upton, Mike McComish, Cillian Willis, Miah Nikora, Shane Monahan.