David and Emma Jackson, of Inholmes Park Road, spotted the iconic Australian bird at about 7.30am near the water tower.
The couple, who live in town with their two children, were both amazed when it flew over their heads and landed in a tree.
"I couldn't believe it," said David, 42, who has previously been to Australia with Emma, 45, where they had seen the birds before.
David, who works as an engineer, said: "I started taking some pictures of it and then it flew off to another tree where it was a bit lower, where I was able to get a better picture.
"Initially I did a FaceTime with my son so I could show him."
David said as soon as he stopped filming the kookaburra swooped down and caught a mouse only a few meters away from him before returning to its branch.
"It ate the mouse and it was quite happy with us there watching," said David, adding that it was only a couple of metres above the ground.
He said: "It was very odd to see it actually hunting in this country.
"I thought if it was raised in captivity it might have lost that instinct to do that but it was pretty deadly."
David said he wanted to hear the kookaburra perform its iconic 'laughing' call but it stayed quiet.
After the encounter he and Emma emailed Tilgate Nature Centre to tell them about the sighting.
Not knowing where the bird could have come from, they also posted a picture on Facebook and asked if anybody had lost it.
David said: "I don't know how it would fare in the winter, I don't know if there would be enough food around for it.
"But it seems to be doing quite well on the outskirts of Burgess Hill. I think it would be quite happy on frogs and mice."
He also said this truly was a chance encounter because he and his wife do not usually go on morning runs but had been building their fitness again after Easter.
"We normally just stay on the pathway but my wife convinced me to go the other way and I'm glad she did."