Edinburgh will also fancy their chances of climbing the RaboDirect PRO12 standings, they are level on points with Elwood's men after winning their last two league games against Ulster and Zebre.
But Connacht, who will be hoping either Leinster win the Amlin Challenge Cup or Munster succeed in the Heineken Cup to hand Ireland a fourth qualifying place, are looking to end their campaign on a high.
"We still have a hell of a lot to play for in those four games," said coach Elwood. "There is a lot of things still in our control.
"We are still looking for that improvement and that betterment year on year and that's very much a target. It starts this weekend now with Edinburgh."
There is only one change to Connacht's starting XV that was named to face Treviso in the last round - Michael Swift returns from injury and takes his place in the second row.
Edinburgh, meanwhile, have named the same back division that defeated Zebre 9-7 last time out but have made three changes to the pack, two of which are enforced.
Hamish Watson, Geoff Cross and Grant Gilchrist replace Ross Rennie, Willem Nel and Perry Parker, who starts on the bench, with interim coach Stevie Scott looking to make it three RaboDirect PRO12 wins in a row since taking charge last month.
"I've always said that that there are a lot of good players at this club so, for us, it's about getting consistency," said Scott.
"That's the challenge of tomorrow night. We need to back up what we did against Ulster, where we saw a good response to this new coaching team, and also from the Zebre match, where we showed a lot of guts to grind out the win.
"We've worked hard and played a lot of good rugby in these games, and have been building a lot of spirit in the squad. We now need to take that forward.
"We need to treat our home games as a big advantage to us. We've the backing of our home fans, it's a big pitch and the ground is quick and hard, which suits our players."
At Murrayfield, 7.30pm.
Referee: Rhys Thomas (WRU, 1st competition game)
Assistant Referees: Peter Allan, TBC (both SRU)
Citing Commissioner: John Kirk (SRU)