After falling behind in the second half, Smith equalised for the visitors but neither side could then go on and find a winning goal.
Wes Foderingham, returning to the ground where he had been on loan earlier in his career, was called in to action early on keeping out a low drive.
At the other end, Michael Smith teed up Yaser Kasim on the edge of the area but the midfielder saw his effort well held by Seb Brown in the Bromley goal.
Smith himself fired wide before watching as his thunderbolt of a shot was beaten away by the Bromley 'keeper.
After the break, the home side took the lead. A mistake in the Town defence allowed Higgins the opportunity to drill a low shot across Foderingham and into the far corner of the net.
Despite the set-back, Swindon were level minutes later. Some neat build-up play between Massimo Luongo and Kasim ended with the latter squaring the ball for Michael Smith to tap home the equaliser.
Nathan Byrne then lashed an effort over the crossbar before the home side went back on the offensive.
Numerous chances came and went and Swindon had Foderingham to thank for keeping the score level combined with the hosts' failure to convert the opportunities they created.
Next up for Mark Cooper's men is a friendly against Leeds United at The CG on Tuesday evening.