Swindon striker James Collins had a close-range shot saved by keeper Paul Jones and the former Peterborough shotstopper also kept out an Andy Williams header before Crawley broke the deadlock from the spot mid-way through the first half.
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.
Debutant Jamie Proctor should have doubled Crawley's lead in first-half stoppage time when put clean through by Clarke, but keeper Wes Foderingham dived to his left to save.
Crawley threatened soon after the interval when a low goalbound shot from Josh Simpson was taken by Foderingham.
Swindon then levelled
through a powerfully struck penalty from Rooney after he was pushed by Joe Walsh.
Crawley keeper Jones brilliantly denied Rooney a second as the substitute came close to giving his side all three points with a near-post header.
Williams should have hit the target for Swindon 15 minutes from time when he wastefully headed wide from a Gary Roberts cross.