The Edinburgh outfit picked up their first victory of the season at Easter Road - and only their second three points at home this calendar year - when they defeated St Mirren at the weekend.
And Fenlon would love to give their fans something to cheer about again as Hibs aim for a second win in a row in Leith.
He said: "It is difficult to win games at home in this league so we have to try to make sure we build that little bit of momentum after picking up our first three points here.
"We've got a cup game to look forward to now and it gets us into the quarter-finals as well, which is nice.
"The confidence is building. We've had a couple of good wins, on the road at St Johnstone and then at home to St Mirren.
"We are unbeaten in a few games now and that gives everybody a lift and there's a decent atmosphere around the place at the moment."
Fenlon says an extended showing in the competition this year is important after Hibs crashed out to Queen of the South at the first hurdle last season.
He said: "The runs in the Scottish Cup in the last two years have helped from a financial point of view.
"We were disappointed to get knocked out of the League Cup early last year so we'd love to go on a little run.
"You have to earn the right to do that. If you go through, there are the two aspects.
"It's another game won, which helps confidence, and it's another game to look forward to financially."
Stranraer dumped Ross County in the last round and Fenlon knows the Scottish League One side are not to be underestimated.
He said: "They had a fantastic result and scored three goals against a Premier League team, which is no mean feat.
"We've watched three of their games and plenty of DVDs and they are a good side.
"They've got some good players who work tremendously hard for each other.
"We know we are going to have to play well. They come here with nothing to lose and we are going to have to play well to get through."