This form means that Glasgow lead the way in the league, one point ahead of Ulster, although they face a Scarlets with plenty to fight for themselves.
The Welsh region are currently just outside of the top four only on points difference and, with three games remaining, will be doing all they can to book themselves a semi-final berth.
And with Scarlets leaving Scotstoun with an 18-13 victory in just the second game of the season, Horne knows full well that the Warriors will have a fight on their hands as they bid to take another step towards a first-ever home play-off semi-final.
"Our form of late has been very consistent, our only defeat in the last nine PRO12 games came at Leinster when the ref called my pass forward at the death and we missed out on a match-winning try," Horne told the Glasgow Evening Times.
"But at the same time, we know our high standards that we have set ourselves in these games can't fall away. Scarlets came to Scotstoun in the second game of the league season and we didn't get to grips with them at the breakdown.
"To be honest, we were on the back foot for the full game. With the likes of winger George North coming on to quality ball, you are always going to be under pressure.
"So we know that we must get off to a very positive and solid start and really try and take the fire out of their bellies.
"If we can get the win down there it really would give us a fantastic platform into the Ospreys game.
"But we know that we need to make sure we get every aspect of our performance right. We believe in ourselves."