It was a scrappy match at a rain-lashed Murrayfield
Edinburgh 14 (6)
Pens: Paterson 2
Glasgow 9 (0)
Pens: Parks 2
Hugo Southwell came off the bench to rescue Edinburgh, who had been in danger of being mugged by Glasgow.
However, the full-back's match-winning try was tinged with controversy as the ball seemed to squirm from his grasp before it was grounded.
The hosts led at the break thanks to two penalties from Chris Paterson but Glasgow replied with two penalties from Dan Parks and a Colin Gregor drop-goal.
Southwell's touchdown was followed by a late Duncan Hodge drop-goal.
The Edinburgh win at a rain-lashed Murrayfield was marred by an injury to breakaway forward Ally Hogg.
The Scotland international was taken off in agony with an injured arm during the second half - and could face a race to be fit for the start of their Heineken Cup campaign next month.
Edinburgh move up to third in the table after a scrappy game, while Glasgow have won just one of their opening five games.
The hosts edged in front in the 15th minute thanks to a 40-metre penalty by skipper Paterson when the Glasgow back line sneaked offside in defence.
Glasgow's Thom Evans thought he had scored a try but Gregor was guilty of a forward pass in the build up.
Phil Godman sent a drop-goal attempt narrowly wide of the target before Paterson banged over penalty number.
The visitors started strongly after the interval and within 10 minutes they were on level terms.
Gregor produced a smart drop-goal following a powerful free-kick surge by Johnny Beattie.
Then substitute full-back Dan Parks slotted a penalty when Edinburgh were guilty of deliberately slowing the ball down in the wake of a great run by Andy Henderson.
Parks repeated the feat after Hodge had obstructed Gregor - only for Edinburgh to seize back the initiative with Southwell's dubious try.
And Hodge completed the victory by clipping over a drop-goal in stoppage time.
Edinburgh: Cairns, Paterson, Di Rollo, Dewey, Webster, Godman,
Blair, Jacobsen, Hall, Dickinson, Duley, Murray, Strokosch, Rennie, Hogg.
Replacements: Southwell for Cairns (40), Hodge for Godman (54),
Hewett for Jacobsen (54), Turnbull for Duley (30), Callam for Hogg (59).
Not Used: Kelly, Laidlaw.
Glasgow: Marsden, O'Hare, Morrison, Henderson, Evans, Gregor,
Beveridge, Va'a, Thomson, Murray, Newman, Kellock, Wilson, Barclay, Beattie.
Replacements: Parks for Marsden (40), Lawson for Thomson (58), Tckachuk for Murray (69). Not Used: Turner, MacFadyen, Barrow, Addison.
Referee: Malcolm Changleng (Scotland).