Felipe Contepomi was able to add the conversion to make it a two-point game but the Argentinian, when presented with an injury-time penalty kick, was off target and the Scottish side held on for the win.
Speaking afterwards, Leinster coach Michael Cheika said: "We hung tough and we showed a lot of determination to get back into the game, but we got sucked into the width of the game and didn't play smart enough.
"It was a scrappy game and obviously we're disappointed because we had a chance to nick it at the end, but we have to take it on the chin and work hard (back in training)."
With Cardiff beating Ulster at home, Leinster's lead at the league's summit is down to nine points with two rounds of matches remaining.
Cheika's side are at home to the lowly Newport Gwent Dragons next time out on May 3.
A win over the Dragons would see them crowned champions and the Australian is not too downbeat despite Leinster's ten-match winning run coming to an end against Edinburgh.
"There's no panic, but we have to look at our performance and learn from our mistakes. Some points (of our play) were quite good. We had a lot of possession, but the problem was that we didn't use it," he admitted.
"We're not presumptuous. We know how hard it is to win on the road and you have to play really well to win away from home in this competition."
Meanwhile, Edinburgh coach Andy Robinson was delighted with his side's display particularly as they bounced back after losing their 1872 Challenge Cup clash with Glasgow Warriors recently.
"If you look at our performances this year then the times when we have done well is when we have only conceded one try.
"Again against Leinster, we only conceded one try and that is the level of performance we have got to reach every time we play," said the former England boss.
"The challenge is always about how a team picks themselves up after a set-back and what pleased me was how well the team responded.
"Leinster are a quality side and they deserve to be where they are - they are the most consistent side in the Magners League this year.
"But there is something about Edinburgh playing Leinster and that was summed up by the way the midfield played. I thought (Phil) Godman, (Nick) De Luca and (Ben) Cairns really took a step forward with their performances.
"As a pack, defending on the line as they did enabled us to control the forward rushes."
Incredibly, Leinster have only won once at Murrayfield in all their visits there. They were 31-8 winners in a league match at the home of Scottish rugby in May 2006.
Their run of defeats at the venue has included two Heineken Cup pool losses (in October 2006 and last December).