Edinburgh open their Heineken Cup campaign with a trip to Reading on 12 November to face London Irish, while Glasgow are at home to Bath.
"We are in a really competitive pool and it is always positive to have a home match first," said Glasgow head coach Sean Lineen.
"This will help us get the players and our fabulous supporters excited.
"We have played Bath before and know how strong they will be but I know we have the squad to win these matches."
Edinburgh welcome Racing Metro 92 to Murrayfield on 18 November and play a double-header with Cardiff Blues in December.
Glasgow open on Sunday 13 November and visit Leinster the following week before home and away clashes with Montpellier.
Rounds five and six will be played in mid-January.
"The sides that we've been drawn against all present winnable opportunities for us," said Edinburgh head coach Michael Bradley when the pools were decided on 7 June.