With Kelty held to a 1-1 draw at home to Elgin City – Kallum Higginbotham grabbing an equaliser in the third minute of stoppage time for Kevin Thomson's side – City now have their sights on the teams above them rather than those lurking over their shoulders.
The Capital side remain fifth, but have won three of their last four League Two fixtures following a disappointing start to the campaign.
Manager Gary Naysmith made five changes to the side that defeated Elgin 2-0 prior to the international break with the former Hearts and Everton full-back only a ble to name five substitutes for the trip south with several key players ruled out through injury, captain Robbie McIntyre and midfielder Danny Handling among those absent.
The visitors' goal was almost breached inside two minutes when Craig Ross saw his header hit the top of Brian Schwake's crossbar following an Anton Brady corner.
The hosts continued to pressurise the City goal, Matty Yates finding space in the penalty area and calling goalkeeper Schwake into action, the American doing well to palm his opponent's effort away.
See then raced through on goal in what was a lively opening ten minutes only to be denied by Luke Scullion in the Stranraer goal.
The home side cranked up the pressure with both Grant Gallagher and Ross Smith seeing efforts flash wide before a Callum Tapping error led to a stramash in the six-yard box.
Schwake pulled off a tremendous double save three minutes before the half-time interval as the 20-year-old denied both Brady and Broque Watson to send the sides in goalless at the break.
Callum Crane spurned a great chance to put City in front midway through the second half when he slipped beyond a static defence only to be thwarted by Scullion.
However, striker See was on hand to fire home the winner with little more than ten minutes to play when he connected with former Blues midfielder James Hilton's cross to secure a huge three points.