They sit one place and six points above their hosts in Pool 3 with two rounds of action remaining.
An impressive run of form has seen them go unbeaten since September, but Easterby knows his side need to be on form in Glasgow to come away with the victory.
"We are doing okay at the moment, but we are not getting ahead of ourselves and we know that we have got a massive game this weekend," he said.
"No-one is pretending that we are playing perfect rugby but we are winning games.
"In big European games nothing but your best is going to be good enough and its going to be a huge game for us at Firhill.
"It's going to be a big week. We have a couple of injuries to look at, Johnny (Sexton) had a tight hamstring and (Gordon) D'Arcy got a bang on the ribs but they should both be fine.
"Glasgow are playing well at the moment and are playing with a lot of confidence so it's going to be a huge game."