Batman was scanning the streets with his night vision equipment (or whatever it is exactly he uses) and found a man in a dark-red costume effortlessly defeating and killing a small street gang. He followed on the roof tops and landed (he thought) silently behind the man when he went into an alley. Of course, Daredevil could hear 5 times better than anyone else and chuckled at the noise behind him.
"Think you're sneaky, huh?" He asked turning around.
"Not bad, the way you handled that gang back there, but killing isn't exactly what we do." Batman said.
"Not what you do." Daredevil corrected.
Angered, batman charged as a heavy rain began to fall.
Background music: Hero by Skillet
Daredevil pulled out his billy clubs, now being able to "see" Batman, due to the rain. Batman rolled forward, now right beside Daredevil as he uncurled and tried to throw a punch with his right fist, but Daredevil did an outside block, moving Batman's arm out of the way (which connected with a building) then Daredevil smacked him across the face with one of the clubs, before jumping backwards and flipping in the air, landing about 5 feet away. Batman threw the same thing he threw at Bane in their first fight in The Dark Knight Rises (I couldn't tell what he threw all I saw was a bunch of flashes and smoke) to attempt to blind Daredevil, but since he was already blind and not using his eyes, this did nothing.
Daredevil shot his grappling hook towards batman, but he caught it and pulled him inward. Batman managed to get a kick in as Daredevil was sailing towards him, knocking him to the ground and tried to stomp on him while he was down, but Daredevil rolled out of the way and connected his billy clubs, making a staff. Batman got tired to throw a punch, but by blocking with his staff and spinning, Daredevil was able to make enough momentum to make Batman keep going while getting behind him, at which point he slammed one of the ends of the staff into Batman's back, sending him to the ground again.
"You know, I'm surprised. I heard you were able to take down whole groups of gangs and terrorists, but you're not doing so great." Daredevil said.
"I haven't started yet." Batman said, throwing a flashbang (or item like it) at Daredevil's feet, going off and temporarily deafening Daredevil, bringing him to his knees.
Batman ran at Daredevil, but he stepped in a giant puddle on the way, making a loud splash that countered the noise of the flashbang, snapping Daredevil back into the fight, looking up just as Batman had jumped towards a building, then pushed off and was about to blast him in the face with a powerful right hook. Daredevil managed to move his head out of the way in the last minute and drove his left his into Batman's stomach, knocking the wind out of him, then did a spinning kick and kicked Batman in the back of the head and knocking his mask off in the process, throwing him to the ground.
End music
He was about to deliver the final blow, bashing Batman's nose into his brain, but then, due to the heavy rain, he was able to "see" who he was really dealing with, one of the people who he represented and was a lawyer for his company:
"Bruce Wayne?" Daredevil said, no longer using his tough, disguised voice.
"You know who I am? Who are you?" Batman asked.
"It's me. Matt Murdock. Your lawyer." Daredevil said.
"Matt?! But, you're blind!" Bruce said, shocked.
"People can be full of surprises." Matt said.
"That's true for both of us. I suppose we can let this go. You need my money and I need you to keep law suits off of Wayne Enterprises." He explained.
"Sounds fine by me. But don't expect me to change my policy too much." Matt said, shooting his grappling hook, zipping onto a building rushing off.
Reasons for winning:
1: Batman's gadgets are mostly to throw off people's visions or involve the opponent seeing them. Daredevil is blind, so vision based attacks won't effect him.
2: Even if his hearing is disturbed, the smallest noise can remind him what's going on and drowns out the previous noise.
3: As silent as Batman is, he still makes SOME noise, it's unavoidable, plus it was raining, which allowed Daredevil to "see" with his "radar" hearing. But, Daredevil can hear even the smallest sounds, so he could hear Batman's silent approach.
4: I did want to make the fight last a little longer, but I couldn't think of enough material, but as to the hand-to-hand combat, Daredevil was trained by Stick, also a blind man but he can take down anyone or do almost anything, such as effortlessly pinning down Elektra, master assassin with deadly strength, so Daredevil's martial arts skills match, if not surpass, Batman's. And they're basically matched in strength.
5: If Kingpin could throw Daredevil around with his grappling hook, Batman, who trains vigorously for everything and is most likely stronger, definitely could.
6: This may be an arguable factor, but since Batman's suit is metal, it may weigh him down more than Daredevil's leather suit, therefore giving Daredevil the advantage of being lighter, faster and more maneuverable.
7: Daredevil is more durable than Batman, as seen where he can be stabbed through the shoulder and still kick ass, whereas Batman can be stabbed in this side, presumably without hitting important stuff and STILL basically gets put out of the fight.
8: Batman's style is slow and analytic, which is trumped by a style that moves faster than Batman could come up with a plan to beat, Daredevil's style is both strategic and fast, making it harder for Batman to plan against him and easier for Daredevil to outdo him.