The Ayrshire side claimed their first victory of the season last week against Ross County to ease the pressure following a poor start to the campaign.
They now face an encounter on Saturday with bottom side Hearts, who will arrive at Rugby Park aiming to crank up the heat again on their 11th-placed hosts.
Barr said: "We thoroughly deserved the win, we played well and everything just came together.
"Hopefully we can kick on now, that's what we have to do. We have to try to build on that result, that's the main thing.
"It's going to be a massive game on Saturday, especially with the predicament that Hearts have been in as well."
Hearts began the season with a 15-point deduction after going into administration.
They head into the weekend action on minus seven points, with Kilmarnock on six and knowing they could stretch the gap to 16 points.
Barr said: "It would be massive for us. We just want to build on something. St Mirren have kicked on and we just want to try to progress as well.
"We had been playing well in spells and it just hadn't been coming together until last week.
"You try not to look at the table and look around you. You know the pressure mounts, but the boys have stuck together and hopefully we can kick on again on Saturday.
"I just look around the changing room and think we've got a lot of experience there and a lot of good young players coming through as well.
"Hopefully that can blend together and we can keep going."
Rabiu Ibrahim is not expected to feature in Saturday's clash.
The midfielder collapsed during last week's win over Ross County and the club are awaiting advice from a specialist after the player spent two nights in hospital undergoing tests.
Better news for Killie is the return of Sammy Clingan, who missed out with a knock last weekend.
Hearts midfielder Callum Tapping claims no-one can afford to write off his side despite their predicament at the foot of the Scottish Premiership.
Gary Locke's side have taken only one point from six games but they previously took seven off their 15-point deduction in their opening four games of the campaign.
And Tapping is convinced they can get back to that form.
The former Tottenham youth said: "I think it's inevitable that a lot of people will write us off considering the point gap but with the quality we've got here, I don't think we can be written off.
"We are a good team and we showed that at the start of the season. We started off well, won a couple of games and once we get that run going again we will be fine."
Hearts' situation has been compounded by Kilmarnock and St Mirren finding form with both sides recording wins last Saturday as Motherwell came from behind to beat Tapping's team.
"It's a big game because of the way results went at the weekend," he said. "We need to win to cut the gap.
"Once we get a win, the boys' morale will be flying, like it was at the start of the season.
"After the first two wins, everyone was buzzing. We started off well and mistakes cost us goals. We still have a lot of games left, we are looking at every game as a positive and another possible three points.
"It's been a big point among the squad that the confidence can't get down, we have to keep our heads up at all times."
Locke hopes to have both Kevin McHattie and Callum Paterson fit and available.
McHattie missed last weekend's defeat by Motherwell through illness while Paterson limped off with an ankle injury.
But both players returned to training on Thursday and are expected to be available, although Billy King is set to miss out again with a foot problem.