Gary Naysmith’s side were boosted by an excellent showing from John Robertson. The former St Johnstone youngster netted twice in the opening period before laying off for James Hilton to secure the points in the dying seconds.
The Blues will feel hard done for their efforts as they dominated much of the second period after Matthew Yates had pulled a goal back, but a combination of some stellar City defending and wasteful finishing meant they left with nothing.
Robertson could've had a first-half hat-trick with better luck. His celebrations on nine minutes were killed by the linesman’s flag after nodding home Luke Berry’s cross.
When he did open the scoring it was in spectacular fashion. Charging in from the right, he aimed for the far corner and found it with an arrowed strike from 25 yards.
Edinburgh's early pressure had already dissipated as Stranraer grew into proceedings following a lethargic start. Craig Ross claimed he was impeded by Jonny Jarron as he attacked a corner at the back post before Yates, who took that set-piece, also found his route to goal blocked by Jarron with a last-ditch tackle.
City gave themselves a cushion going into half-time when Robertson headed in off the crossbar from Hibs loanee Innes Murray's cross.
After manager Jamie Hamill, once of Hearts, introduced two substitutes at the break and changed his side into a 4-4-2, Yates took advantage of a mix-up in the City defence to run through on Schwake before passing into the bottom corner.
That was almost wiped out immediately as Ouzy See broke into the visiting box and was hauled down by Ayrton Sonkur. Hamill's former Tynecastle team-mate Callum Tapping couldn't take advantage though, seeing his low effort pushed away by Luke Scullion.
Yates almost had his side level in spectacular fashion as the away winger embarked on a mazy run down the right and fired a strike which narrowly missed the far post. The attacker will be more disappointed with a late chance he passed up by side-footing straight at Schwake.
It was then left to Hilton to wrap things up for City. The substitute kept his cool after receiving from Robertson, slotting into the far corner.