Half-time substitute Borja was denied from the spot by birthday boy Danny Vukovic and hit the post twice – the second coming in the last minute of stoppage time – as Colombia failed to show the clinical edge that saw them come from two goals down to beat Les Blues 3-2 in their previous outing.
Jose Pekerman's team were by far the better side at Fulham's Craven Cottage and produced plenty of the fast attacking breaks and skilful forward play they hope will bring them success at the World Cup, but ultimately failed to convert a number of excellent chances.
The best of those fell to Palmeiras forward Borja, who was afforded frequent opportunities by a ragged Australian defence that had four goals put past it by Norway last time out.
On a frustrating night for Colombia, Pekerman also saw defender Oscar Murillo carried off with a potentially serious injury.
The result and a clean sheet will come as a relief to Australia coach Bert van Marwijk who, while failing to claim his first victory since taking the job, will have been pleased to earn a respectable draw.
Colombia's first opportunity came in the fifth minute. Socceroos goalkeeper Brad Jones was fortunate not to be punished when he flew out of his goal to punch the ball clear and was beaten to it by Andres Uribe, who headed narrowly wide of the unguarded net.
James Rodriguez carved out a one-on-one with Jones five minutes before half-time but failed to get any real power behind his shot, before Tomi Juric dragged one across the face of goal and narrowly wide at the other end.
Neat interplay between Tom Rogic and Massimo Luongo early in the second half allowed the latter to attempt a swerving left-footed shot from 20 yards, but his aim was wayward.
Borja replaced the ineffectual Radamel Falcao at half-time and his first contribution was to pull off an audacious overhead kick that flew narrowly over the crossbar.
However, Colombia were dealt a blow when Murillo suffered a leg injury and had to be taken off on a stretcher shortly before Borja had a goal correctly ruled out for offside.
Colombia continued to pile forward and Johan Mojica's looping cross presented Borja with a free header, but he contrived to bundle wide from inside the six-yard box.
With 10 minutes left and Australia unable to get out of their own half, yet another chance fell to Borja as he nodded across goal and struck the upright.
There was still time for Vukovic - a half-time replacement for Jones - to clumsily fell Borja in the penalty area, but he redeemed himself by diving low to his left to save the ensuing spot-kick in the 86th minute.
Borja's miserable outing was compounded when he hit the post in the last action of the game as Australia narrowly avoided defeat.