The livestock were grazing in a field in the Nether Haugh area of Rotherham, when a sandy-coloured Staffordshire Bull Terrier ran into the field chasing them for around 15 minutes.
A passer-by who filmed a part of the attack, captured the dog hanging off the neck of one of the sheep, who had managed to flee the field in fear.
CCTV footage from a nearby property also caught some of the attack with the dog seen chasing the sheep for around 15 minutes before he backs some of them into a corner, before attacking the other sheep.
Two of the sheep were seriously injured and required emergency veterinary treatment.
Farm owner Nick Grayson said: "The dog has ripped one of the sheep's chin and left her with a hole which means whatever she eats, it just leaks out.
"The other sheep has puncture marks on her face and puncture wounds to her leg.
"All 60 of them were chased around for a good 15 minutes and that would have been really stressful for them.
"We are due to start lambing in a couple of weeks and I just don't know what is going to happen because of the stress of the attack.
"There could be dead lambs inside them. It is just a waiting game at the moment."
Mr Grayson is now appealing to the owner of the dog to come forward following the attack, which is believed to have happened early on Sunday morning.
He said: "It makes me so angry.
"The dog will have gone home covered in blood so it will be pretty obvious that something bad has happened.
"An attack like this is devastating to deal with. It affects everyone on the farm. I have two little boys, the eldest is three-and-a-half and he has definitely picked up that something isn't right."
Both sheep are expected to make a full recovery, but Mr Grayson is keen to avoid a repeat scenario in future.
He has urged dog walkers to keep their pets on a lead, no matter how well trained they are.
He said: "Just, please keep your dog on a lead, it couldn't be any simpler.
"Don't think 'oh, my dog would never do anything like that' because they could. It is a sheep's instinct to run from a predator and it is a dog's instinct to chase pray.
"Even if your dog has the perfect recall, keep it on a lead. If people see your dog off the lead, it just encourages them to do the same and this just increases the risk to livestock."
South Yorkshire Police are also repeating Mr Grayson's calls for dog owners to keep animals on a lead when walking near livestock.
PCSO Jacqui Millward, of South Yorkshire Police, said: "We have been made aware of an incident in the Greasbrough / Nether Haugh area of Rotherham
"A light Sandy Staffy dog has attacked sheep that were grazing in the local farmers field. Sadly two sheep have been seriously injured, requiring veterinary care. We are pleased to say the sheep will make a full recovery.
"Can we please remind all dog owners it’s their responsibility to make sure their dogs are under control, especially vigilant around livestock."
Anyone with information about the attack can call South Yorkshire Police on 101.