The Shrimps’ striker scored a vital leveller nine minutes from time to restore parity after a Bez Lubala’s first-half penalty put Crawley ahead.
It was the fitting reward for a spirited Morecambe display that saw them seal the point in Derek Adams’ first league game in charge.
Morecambe started the game on the front foot and Aaron Wildig missed a great chance when played in by Lewis Allesandra with six minutes played.
Wildig then set-up Stockton for an even better chance with a clever header but the big striker – one-on-one with Morris in the home goal – missed with a shot which just shaved the post.
Crawley showed they had a threat up front themselves however and Lubala forced a good save from Mark Halstead from a free kick and Ashley Nadesan headed wide from Nathan Ferguson’s cross following a corner.
Ferguson himself missed with a shot following another cross by Lubala after 21 minutes before the game hit became something of a scrappy affair. The game looked set to be goal less at the break but in the 43rd minute David Sesay broke into the Morecambe penalty area and was brought down by Tom Brewitt.
Referee Antony Coggins pointed to the spot and Lubala coolly slotted the ball home past Halstead for his ninth goal of the season to give Crawley a boost before the break.
Morecambe hit back and had the first half-chance of the second period when Sutton narrowly missed the target with a header from a corner after 49 minutes.
As the game entered its final stages Adams brought on John O’Sullivan for Steven Old and the change had an almost immediate effect as Morecambe levelled with nine minutes to go.
Brewitt was found down the right by Tanner and fired in a low near post cross which was turned in by Stockton from six yards out.
The home side missed a great chance to win the game when Nathaniel-George’s low cross flew to the back post, where both Dannie Bulman and Grego-Cox left the ball for one another and Halstead was called into action to save well from Nathaniel-George as the Shrimps hung on to seal a well earned draw.
Crawley Town: Morris, Young, Tunnicliffe, Dallison-Lisbon, Sesay, Bulman, Grego-Cox, Camara, Ferguson (rep Bloomfield 83), Lubala (rep Nathaniel-George 61), Nadesan (rep Palmer 69).
Subs not used: Luyambula, Sendles-White, Francomb, Galach.
Morecambe: Halstead, Lavelle, Sutton, Old (rep O’Sullivan 78), Conlan, Kenyon, Wildig (rep Leitch-Smith 63), Tanner, Brewitt, Alessandra, Stockton (rep Ellison 84).
Subs not used: Roche, Miller, Tutte.
Ref: A Coggins.