Stranraer had conceded 10 goals in their opening two games, but looked much more compact and harder to break down as Dundee visited Stair Park.
However, Dundee began knocking at the door in the final few minutes of the half.
Stranraer failed to clear a Niall McGinn corner on the right and when the ball broke to Cammy Kerr on the edge of the box he met it with a first time volley.
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Paul Woods was on the line to head clear and then moments later denied Dundee captain Ryan Sweeney from another set-piece.
Two minutes from half-time the Championship side did break through as McGinn floated in the corner and Kerr’s downwards header squirmed under the body of George O’Connor.
In the second half Alex Jakubiak missed a great chance to double the lead, but a second would arrive on 71 minutes.
Sweeney rose to head the ball back across goal and Paul McGowan was unmarked to smash the ball beyond O’Connor.
The game was wrapped up eight minutes from time as Luke McCowan cut in from the right to curl the ball into the net.
It was a rather more even contest on Tuesday as Stranraer travelled to Station Park to face League Two rivals Forfar,
A Sam Ellis own goal gave the home side the lead going into the break, but James Hilton levelled just three minutes after the restart with a tidy turn and finish.
Defender Kyle Girvan saw red with 15 minutes left for a second bookable offence, and it took just one minute for Forfar to take advantage as Josh Jack swept home from a ball into the box.
Stranraer weren’t done yet though and with four minutes left substitute Scott McLean won a penalty, which he scored himself.
Then in stoppage time Forfar were handed a penalty, and the chance to win it, when Luke Scullion brought down Yusuf Hussain, but Stefan McCluskey put his effort over the bar .
So it went to a penalty shoot-out which Forfar won 5-3 to claim the bonus point after Marc McCallum denied Hilton.