A brace of tries by Richie Vernon and another by Niko Matawalu had put the Warriors within touching distance of their first try-bonus point of the season, but Leinster's defence held firm late in the game and Glasgow had to settle for a losing bonus instead.
Despite the outcome, Gregor Townsend, the Glasgow coach, said the performance of his side had given him hope that they can push on over the coming weeks and seize a semi-final slot for the third consecutive season.
"It was a cracking game with two very good sides going at each other for 80 minutes in difficult conditions and there were some great tries scored," said Townsend. "It's disappointing we didn't come away with a win but we've gone up against the defending champions, really put them under pressure and scored a couple of good tries.
"I was very proud of the players, the way they took the initiative in attack and especially how we defended with real effort going in and some big tackles being made to defend our line. The performance sets us up well with eight games remaining, six of which are at home."
The result means that Glasgow stay in fifth place in the PRO12 table, three points behind fourth-placed Ospreys. However, Townsend's team have two games in hand over most of their rivals following the cancellation of their Scotstoun matches with Treviso and Edinburgh during the festive period.
Warriors officials expect to be able to announce dates for the rescheduled fixtures within the next fortnight. The likelihood is that they will take place on the weekends of the Heineken Cup quarter and semi-finals - April 5/6 and April 26/27 - but that would require dispensation from European Rugby Cup Ltd, whose participation agreement demands that no fixtures should clash with their knockout stages.
Glasgow might also face problems if there is another period of bad weather. The Scotstoun pitch is still looking decidedly threadbare and a combination of heavy use and heavy rain would mean that more cancellations could not be ruled out. That said, Edinburgh's success in taking last Friday's match with the Ospreys to Boroughmuir's Meggetland ground might tempt them to have something similar on standby.
Vernon's first-half brace of tries was another ringing endorsement of his decision, taken with the encouragement of Scotland coach Scott Johnson, to move from the back row to the midfield. Matawalu's touchdown after 50 minutes was a typically crafty piece of work by the Fijian, who deceived the Leinster defence - and referee Nigel Owens - with a low dive around the side of a ruck.
Finn Russell had another successful evening with the boot, converting two of the three Glasgow tries and adding a couple of penalties. Leinster's points had come from tries by Noel Reid, Darragh Fanning and Rhys Ruddock, plus two conversions and three penalties by Ian Madigan. However, the most impressive part of their game was seen in the final 15 minutes when they showed strength and resolve in defence to thwart a powerful finish by Glasgow.