The capital club also won 2-0 against Forfar Athletic and Swift thinks that after another two matches, it could be neck and neck.
“I think it is a stick-on it will go down to the last day, and if it does, anything can happen with the pressure of it all,” he said.
“We would relish it and they would too.
Nicky Hogarth: Former Graeme High pupil signs for Falkirk on loan from Premier League newcomers Nottingham Forest
Falkirk set to bring in goalkeeper ahead of Peterhead match
McGuffie talks style of play and mentality ahead of Peterhead clash
Syngenta: Finlay could be playing 'four or five' levels above the EoS second division, says boss
'Things did conspire against us' - Boss John McGlynn on Falkirk's second-half display
“They have improved recently and have shown character all season long.”
Goals from Adam Corbett and Michael Miller sealed Stenny’s win on Saturday.
Of their display, Swift added: “It was very comfortable for us, and we should have won by more goals, to be honest. We had one chopped off for nothing and we cruised it.
“The opening goal from Adam was what I have him in the team for. He gets up the pitch from full-back well.
"The second was a cross for sure but we’ll take it. We deserved that bit of luck.”
Saturday sees the Warriors face an Albion Rovers side coming into the match on the back of a 5-0 scudding by Stranraer.
Swift thinks this is the worst time to face Brian Reid’s usually-organised outfit, though.
He said: “They play 4-4-2, and normally they do it well and the players work the shape and put in a shift. I was shocked to see that scoreline and it isn’t good news for us.
“When a team loses heavily, the next game they always come out fighting, and I am sure they will be right up for it against us.”