The Scotland Under-21 forward opened the scoring in the 24th minute before Bosnia-Herzegovina defender Sanel Jahic, making his first start since joining the Perth club last month after leaving Turkish club Karabukspor, grabbed a second in the 61st minute from almost on the line.
The Perth side have never won a major trophy but play Aberdeen in the semi-final of the Scottish League Cup in February and will be in the Scottish Cup fifth-round draw on Monday, with May insisting there is belief within the squad that they can make history this term.
"The boys believe it can be done this season and there is no reason why not," he said.
"It is possible this season. Obviously every season you believe, but around the club there is a real buzz for the cup matches.
"The draws have been good for us. We have a semi-final against Aberdeen but we fancy ourselves against anyone, especially in a semi-final.
"We have got a lot of quality in that team and we work as hard as any, so there is no reason we can't get to a final and anything can happen then."
Saints goalkeeper Alan Mannus made three crucial saves to deny Livi in an entertaining, end-to-end cup-tie.
However, Saints boss Tommy Wright, himself a former keeper, feels his team should have made life easier for themselves.
"Alan is a good goalkeeper and he contributed to a good win but I wouldn't say he kept us in it," the Northern Irishman said.
"He made some decent saves but at the other end we were wasteful. We had a lot of good opportunities that we made the wrong decisions.
"We started the game well and then got sloppy after the goal and after the second goal we definitely got a bit sloppy at both ends of the pitch.
"We gave them one or two opportunities. It should have been more comfortable but the main thing is we are in the hat for the draw."