Edinburgh claimed the Cup last season and they have the upper-hand this time around, after John Hardie and Matt Scott crossed over in the capital.
Mark Bennett scored Glasgow's only try and Horne, who partnered him in the centres, admits there is plenty to work on ahead of the return fixture at Scotstoun on Saturday.
"When we looked after the ball, we looked dangerous and caused them problems," Horne told The Courier. "But ultimately our ill-discipline and not being 'ball-tough' enough was the problem.
"We lost the ball in contact countless times, and fair play to them they worked really hard at the breakdown, slowed us down and we struggled to get the tempo into the game that we wanted.
"There were only two times we got above five phases in the first half and Mark's (Bennett) was one of them. If we get tempo in our game and keep our shape we are dangerous and score tries, but we coughed up too much ball.
"We also never really made our points stick; we'd get back within two points and then give away a penalty, they'd kick to the corner, get another penalty and before you know it it's back to five points. Then they kick to the corner, lineout drive and there's the try.
"We just need to play with real intensity to match them in 80 minutes, and it's something we'll need to do on Saturday. But we're confident we can make the adjustments."
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