The completely canon answer
Really this shouldn't have happened in the books he is never actually picked up by the Dementors, the only spell he uses other than Expecto Petronum is Lumos to try and find his wand when it is lost on the floor.
“Where’s - wand -come on -lumos!”
He said the spell automatically, desperate for light to help him in his search - and to his
disbelieving relief, light flared inches from his right hand - the wand tip had ignited.
Harry snatched it up, scrambled to his feet and turned around.
His stomach turned over.
A towering, hooded figure was gliding smoothly towards him, hovering over the ground, no feet
or face visible beneath its robes, sucking on the night as it came.
Stumbling backwards, Harry raised his wand.
“Expecto patronum!”
-- Order of The Phoenix
Also in Prisoner of Azkaban in The Marauders Map chapter Lupin says this
“There are — certain defenses one can use,” said Lupin. “But there was only one Dementor on
the train. The more there are, the more difficult it becomes to resist.”
We never hear of any other defenses than Expecto Petronum and Lupin later goes on to teach harry Expecto Petronum although he does say he is not an expert in fighting Dementors.
Harry had a sudden vision of himself crouching behind a Hagrid-sized figure holding a large
club. Professor Lupin continued, “The Patronus is a kind of positive force, a projection of the
very things that the Dementor feeds upon — hope, happiness, the desire to survive — but it
cannot feel despair, as real humans can, so the Dementors can’t hurt it. But I must warn you,
Harry, that the charm might be too advanced for you. Many qualified wizards have difficulty
with it.”
The Quote above confirms for me that he did not do a short burst/stab of Expecto Petronum has it really doesn't damage the Dementors also as per book canon nothing damages a Dementor as a Dementor cannot be destroyed. Expecto Petronum merely repels the Dementor with positive energy. In canon there is no explanation for how what he did affected the Dementor, there are however plenty of explanations and possibilities for what his wand did or what he made his wand do.
Wizards cast spells when they lose controls of their emotions, Disappearing Glass and Blowing up Aunt Marge.
Non Verbal magic, in the Potterverse a spell can be cast without uttering the words but by thinking of them.
There are other explanations one which would have less foundation would be Harrys' wand being special take for instance when it snapped Voldemort or Lucius' wand in the Deathly Hallows. Harry has always swore blind that it was his wand that did that and Harry had no part in it.
I have however never heard of using a wand as basically a Cattle Prod, the closest thing I would think is casting stupefy at close range.
The movie answer
To make it more dramatic Harry had to fight off the Dementor first and have this struggle. On screen this looks much better than Harry stumbling around on the floor looking for his wand in the dark.
