They trail second placed Cardiff Blues by three points and clash with defending champions and group leaders Toulon in Round 6 but they still have a slim chance of qualifying for an extended run in Europe.
Ruaridh Jackson was the catalyst for that comeback with head coach Gregor Townsend delighted with that historic double and a debut Heineken Cup win on the road in England.
"We might not have done as well as we would have wanted in this season's tournament but it was a great effort to come here and get that double - which was something we had talked about as a goal in our build-up to the match" he said.
"We will have a good look at the rest of the results from the other teams over the next day or two but we knew that if were able to pick up a win here against Exeter, and other teams were to lose, then there could be a chance of getting into the Amlin if we were to pull off a win next week against Toulon.
"However, we didn't play particularly well and both sides made too many mistakes. We defended well, made our tackles and competed at the rucks but the second half did not really get going for either team while we hung on in there.
"The Chiefs played well in what was a better first half when we understandably looked a bit rusty after having had two games called off.
"Now we cannot wait to welcome Heineken Cup champions Toulon for what is probably going to be the biggest game we have had at our new ground.
"When you have Toulon coming and there is nothing to lose, and if we play like we did in the second half against them when the game was a wee bit away, that was some of our best rugby that we have played this year.
"We know how difficult it is going to be because they will be coming to Glasgow looking for a win to get a home quarter-final tie. They have great depth and quality throughout their squad but we believe we can play better."