Sean Higgins put County in front in the seventh minute when he netted from close range, but Stranraer came back strongly and deservedly levelled in the 70th minute when David Hamilton's shot was deflected into the net.
Andy Dowie cut out a dangerous cross from Lionel Djebi-Zadi as Ross County threatened in the first minute.
It was all County in the early stages, and they took the lead when a short corner from Kevin McKinlay was whipped into the box by Don Cowie and Higgins stooped on the six-yard box to head home.
Stranraer's Bryan Gillfillan moved forward in the 14th minute, and he forced a good save from County keeper Stuart Garden.
County should have doubled their lead in the 27th minute when a John Rankin free-kick was cleared as far as Djebi-Zadi. The Frenchman's overhead kick was deflected into the path of Jim Lauchlan, but the County captain's shot flew wide.
Stranraer keeper Barry-John Corr twice raced off his line at the start of the second half as County looked to increase their lead.
The Blues were pushing players forward in an attempt to get back on level terms, and a Lee Sharp cross in the 62nd minute had the County defence in trouble but Murray Henderson's header went high and wide.
The visitors' pressure paid off eight minutes later when a seemingly harmless shot from Hamilton was deflected past Garden off Djebi-Zadi and went in off the inside of the keeper's left-hand post.
Stranraer kept up the pressure, and Hamilton rattled the bar with a header after he got on the end of a Stephen Swift cross.
The away side were well in command, and Moore almost sealed County's fate when he just failed to convert a Henderson cross six minutes from the end.
It was one-way traffic and Rankin had to get back to block a goal-bound effort from Paul Walker before Gilfillan fired wide from an 88th minute free-kick.