Two of the ladies caught up in the viral fight at ritzy Beverly Hills steakhouse Steak 48 are serving up their side of the beef ... explaining exactly how a birthday celebration turned into a throwdown.

The night started innocently enough, the women tell TMZ ... one of them heard men at a nearby table speaking Arabic and struck up a conversation -- and one of the men later came over and offered to cover their bill.

They say another woman was already approaching the table when he made the offer ... and then sat on his lap.

When the men were ready to leave, one asked them to tell the woman to get off his friend's lap. They tell us they passed along the request -- and got a physical response ... claiming the woman came up from behind swinging and pulling hair.

They say that led to a few punches in return ... with her getting "socked" several times.

The women deny anyone jumped her ... noting she was briefly placed in handcuffs that night but ultimately released.

Despite all that, they say they wish the other fighter the best ... just don't expect them to break bread again anytime soon.