GA pilot reported seeing a UAS close by at 2;500 feet. The pilot climbed to avoid the UAS and flew into Class B airspace without clearance.

Narrative

Flying westbound about 6 to 7NM north of ZZZ at 2;500 ft MSL squawking 1200 under the Houston Class B airspace. My plan was to fly at that altitude until clear of the Bravo outer ring and then climb to a higher altitude. With about 1.5 NM before clearing the outer ring of the Bravo; I noticed what appeared to be a drone or possibly a larger bird ahead of me at my 11 o'clock; slightly lower than me and climbing. I started my climb a little earlier than planned thinking I wouldn't climb into the Bravo. The distraction prevented me from adequately monitoring my position and I was in the Bravo for about 1 NM up to about 3;300 ft MSL. Looking back; I could have reset the autopilot target altitude to 2;900 ft instead of the 8;500 ft I had previously set it to as I was preparing to climb in a few moments. A turn would have been an option as well; but there was a Class D to my left so a right turn was the only option. If I were to have made that right turn; I would have lost visual with the potential threat.