Dougie Fife notched Edinburgh's only try and skipper Greig Laidlaw kicked the other points for the home side, while Stuart Hogg bagged a touchdown for Glasgow and Duncan Weir contributed the rest with his boot.
The match rarely sparked into life - a situation that was not helped by the appalling state of the playing surface which again cut up badly as the game progressed.
Glasgow were second best for most of the encounter but earned their success with a solid showing in the final 20 minutes.
Edinburgh were left ruing their inability to kick on after reaching the break well in command.
Indeed, they failed to score in the second period.
Glasgow made a decent start and Weir opened the scoring in two minutes with a 45-metre penalty. But the visitors held that advantage for only five minutes before a collapsed scrum inside the Glasgow 22 allowed Laidlaw to square matters.
Weir restored the three point gap with his second penalty before Edinburgh produced the afternoon's first move of any note with Laidlaw's slick service stretching the Glasgow defence and freeing Jack Cuthbert, whose deft inside pass freed Fife to complete the job. Laidlaw added the extra two points.
Weir trimmed the deficit with penalty number three but Laidlaw cancelled that out with a long-range effort after an offence that earned Glasgow flanker Tyrone Holmes a yellow card.
The temperature rose sharply in the closing minutes of the first half and Glasgow were reduced to 13 men when Hogg was despatched to the sin bin following an altercation with Edinburgh winger Tom Brown, who was also sent for 10 minutes on the sidelines.
With Holmes preparing to return to the fray, Cuthbert butchered a gilt-edged scoring opportunity. However, the referee was playing advantage and the penalty handed Laidlaw the final scoring opportunity of the opening 40 minutes, and he duly steered the kick between the sticks to send the hosts in at the break with a 16-9 advantage.
Edinburgh restarted strongly and Dave Denton was thwarted just short as he hurtled towards the line. Glasgow coach Gregor Townsend rang the changes and that provided some momentum for the visitors who cut the gap to four points when Weir booted another penalty.
Laidlaw recorded his only miss of the match a couple of minutes later and Weir ensured a lively finale when he banged over another long-range penalty as the game entered the final quarter.
The game swung in Glasgow's favour with 13 minutes to play when they broke at pace and the ball was transferred to DTH Van Der Merwe who chipped ahead and Hogg won the race to the line.
Weir was off target with the conversion attempt but was instrumental in a solid defensive effort as Glasgow saw out the remaining minutes to chalk up the victory.