Both sets of supporters were frustrated in the early stages of the match as neither side was able to take full control of proceedings.
However, a quiet opening period was followed by an entertaining exchange as Wolves took the lead in the 25th minute when James Henry picked out the top corner of the goal with a long-range effort.
The visiting side would only be ahead for two minutes before Crawley hit back through Billy Clarke, who took advantage of a lack of communication at the back from Wolves to convert the equaliser.
Wolves should have been level before the break but Henry failed to find the net as he did earlier in the game, with his shot from inside the box flying over the crossbar.
It was Henry who had the first real opportunity of the second half when he saw his curled effort pushed away by Paul Jones.
Wolves enjoyed most of the possession in the second half and continued to threaten, with Bakary Sako being denied by Jones after trying his luck from distance.
Crawley defended their lead admirably and were rewarded with all three points as Wolves failed to turn pressure into goals in the latter stages of the match.