A goal in each half put paid to any chances Argyle had of getting something out of the National League Southern Premier Division match in Sussex.
Crawley started well with Faye Robson striking the crossbar before debutant goalkeeper Rebekah Kyle saved well from Megan Stow from the resultant corner
Argyle had to wait until the eighth minute to have their first chance when Jessie Jones took advantage of a defensive mix up but curled her effort just over the crossbar.
The Pilgrims were hit in the 26th minute when good work by new Crawley signing Tash Stephens ended with Sam Pittuck curling a superb shot into the top corner of the net.
Argyle gradually grew into the game with Crawley goalkeeper Frankie Gibbs twice saving well.
The visitors had a scare early in the second half when Pittuck beat Kyle to the ball but a strong challenge made Pittuck’s effort strike the outside of a post. Helen Bleazard then passed up a great opportunity to make it 1-1.
Three minutes later Pittuck had better luck when good work between her and Owen ended with Tash Stephens curling in another great strike to make it 2-0.
Argyle responded with a good shot from distance from Kayley Lane.
The Pilgrims were looking much the stronger in the last 20 minutes and brought on 16-year-old Beth Everson for her senior debut by replacing Amber Pollock.
With 17 minutes to go striker Natasha Knapman was denied reducing the arrears when Gibbs blocked her effort. Two minutes later Gibbs made another good save when diving low to deny Jones.
Argyle were given a lifeline seven minutes from time when Stow was shown a straight red for an alleged stamp on Faye Ivall. Ellie Bishop came on for her debut and Argyle kept pushing forward but to no avail.
“I think Crawley will be in the top three,” said Leonard afterwards. “They scored two goals from outside the penalty box which were superb.
“The biggest disappointment is we neglected runners which gave them the opportunity to shoot from outside our box.
“We were better organised and fitter than them despite them throwing everything at us in the opening half an hour. We weathered the storm only for them to score a wonder goal.”
He added: “’I was delighted at the way we played in the second half – the way we pressed them. We pushed and pushed but to no avail. We had chances but they had two half chances and scored.
Argyle: Becky Kyle, Leah Burridge, Katie Middleton, Mollie Taylor, Faye Ivall, Zoe Cunningham, Ellie Bishop, Becky Dandridge, Kayley Lane, Helen Bleazard, Natasha Knapman, Ebony Dover, Amber Pollock, Beth Everson, Jessie Jones.
Meanwhile, a very young Plymouth Argyle development side went down 9-0 at Callington Town in their latest pre-season friendly.