Striker Billy Clarke stepped up to score a left-footed penalty into the middle of the goal after Nicky Adams was tripped by Alan McCormack.
Swindon earned a point with a spot-kick of their own which substitute Adam Rooney converted.
He said: "It was a good result and keeps the scoreboard ticking over. Swindon had beaten us by 3-0 in each of our last three meetings, so this performance showed how far we've come."
Barker, who gave full debuts to Mustapha Dumbuya, Matt Sparrow and Jamie Proctor, was especially pleased as seven of his team were under the age of 21.
He added: "I thought our players were fantastic on a pitch that now looks like a ploughed field.
"Both penalties were a bit soft, but it was a bit difficult to stay upright on the pitch. How often do referees even things up?
"Swindon are a big, strong side and our players are learning and growing on a weekly basis.We had to weather the storm early on but we more than played our part in a good display and a good point.
"This group are young and they will get better and it has started.We have produced some good players at Crawley that have served the club well and the process is continuing."