To the best of my knowledge, this page accurately describes the procedure that I followed to remove the engine computer from my 2000 A4 1.8TQ. I cannot confirm that this procedure will work for your particular vehicle.
Use these instructions at your own risk.

• Ratchet, 2 inch extension, and 8mm socket for unbolting ECU cover.
• Flat-head screwdriver for prying off ECU clip.
   

Click on the accompanying pictures to see a high resolution version.
 

1. Locate the ECU.

Yellow arrow points to ECU Pop the hood. The ECU is located under the black plastic cover on the driver's side, just below the wiper arm joint.

2. Remove protective cover.

Red arrow points to weatherstripping Push the rubber weatherstripping towards the front of the car to free the front edge of the cover. The weatherstripping slides off quite easily. Lift plastic cover Lift the front edge of the plastic cover and slide the rear edge away from the windshield. You need not remove the entire cover. Just lift it high enough to prop it against the hood of the car.

3. Unbolt lid.

Lid held by 5 hex bolts The ECU is protected by a plastic black box. To access the ECU, you need to remove the lid of the black box. The lid is held by five 5/16" (8mm) hex bolts. The first 3 bolts (red circles) are easy to remove since they are not obstructed. The 4th bolt (left yellow circle) is partially obstructed by the plastic trim at the bottom of the windshield, but can be reached using a pair of pliers or a ratchet.

4. Unbolt hidden bolt (if required).

5th bolt The 5th bolt (red circle) is difficult to remove because it is obscured by the trim on the bottom edge of the windshield. It is not necessary to remove the 5th bolt. You can just lift the front edge of the black box cover as shown below. This gives you just enough space to pry on the ECU clip and slide the unit out from the black box. Wipe away any water around the box before removing the lid. Lifting lid with 5th bolt in place To remove the 5th bolt, first turn the ignition to the on position, turn the wipers on and switch the ignition off when the wipers are in the vertical position. Remove the silver clip holding the windshield trim. This allows you to lift the trim giving you a little extra room to work on the 5th bolt. Be careful lifting the trip as too much force may crack the trim or windshield. Silver tab holding windshield trim Reach into the gap between the black box cover and windshield trim with the ratchet+extension, and loosen the bolt, careful not to drop the bolt into the crevice.

5. Unclip ECU.

Red arrow points to end of clip The ECU (silver box) is held to the bottom of the black box by a clip. Release the ECU by prying on the tabs at the end of the clip.

6. Unplug ECU.

Two electrical harnesses are attached to the back of the ECU. Remove them by pulling on the black tab and then pulling the connector straight away from the ECU, careful not to bend any pins while doing this.