The 26-year-old, who has 11 caps, has been on the bench for his club for much of this season.
But he will start ahead of Scotland's four other strikers.
"Am I a bit rusty right now? Yeah, but as a striker you need to be ready whether you're starting the game or coming off the bench," said Griffiths.
"The gaffer has kept us on our toes this week with the tempo of training."
Griffiths scored 40 goals for Celtic last season but was given only 15 minutes of action during the whole of Scotland's failed Euro 2016 qualifying campaign.
He had found Sheffield Wednesday's Steven Fletcher, Norwich City's Steven Naismith and Chris Martin, currently on loan at Fulham from Derby County, chosen ahead of him by head coach Gordon Strachan.
And Fletcher's clubmate, Jordan Rhodes, is the only player to be playing regularly for his club at present.
But Griffiths has still managed 14 goals in 33 appearances for Celtic this season despite finding himself second choice behind summer signing Moussa Dembele.
He was given half an hour along with Rhodes as replacements for Naismith and Martin in Wednesday's 1-1 friendly draw with Canada.
Strachan admits that, with the Scots fifth in qualifying Group F, his side must beat the second-placed Slovenians to retain hopes of reaching the 2018 World Cup finals in Russia.
Griffiths feels he can cope with such pressure despite admitting to nerves as he looks to score his first international goal.
"Maybe I am trying too hard and I just need to relax a little bit more," he said ahead of Sunday evening's game at Hampden.
"I want to try and break my duck for Scotland as quickly as possible.
"It's frustrating for a striker, not scoring, but it's not about me - it's about the team and how I can help them to get the three points."
Strachan's future is on the line should his strategy fail, but Griffiths insists the squad is fully behind the former Celtic manager.
"There's a lot of things said about the gaffer that is probably unfair," said the striker. "We as players need to take on that criticism as well.
"Everybody loves coming away with Scotland and playing for the gaffer.
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"I've seen things with people saying he should resign and this and that.
"But us as players, we want to get a good win and three points, which would put us just a point behind Slovenia with the England game coming up in the summer.
"We still feel we're going in the right direction under the gaffer.
"Everyone is still playing for the same cause. We want to get to the World Cup as much as anybody.
"The results recently haven't been good, but one win changes everything.
"In football, you always need a bit of luck, but we have got enough in this squad to get the win."
Celtic striker Leigh Griffiths has yet to score for Scotland but is ready to lead the line against Slovenia despite admitting he is "rusty".