Graham Lockhart, 23, and Shauna Robb, 20, from Bathgate, West Lothian, were killed in the crash along with their children, Tyler David, aged two-and-a-half, and Arianna Rose Sheryl, aged 11 months.
The family were killed when their Fiat was in collision with an articulated lorry on the A801 near Polmont, near Falkirk on Wednesday.
In a statement released through Police Scotland, relatives of Mr Lockhart said: "The Lockhart family are devastated at the loss of their son, daughter-in-law and two grandchildren, and wish to be left to grieve in peace."
Ms Robb's family released a separate statement which said: "The Robb family are devastated at the loss of their daughter, her partner and their two grandchildren, and wish to be left alone to grieve."
Tributes were also paid to the family on social media sites.
Sheryl Jones, Mr Lockhart's sister, said: "To have you ripped away from us is hard enough, but to have all four of you torn away is just crushing. Never felt such pain baby bro.
"I love the four of you with all my heart. My only comfort is you are all together.
"Wish use (sic) would all just walk through the door and this to be a horrible nightmare. You'll never understand how much we all loved you four."
Joanne Pirrie wrote: "Look after each other up there. R.I.P all four angels. Gone too soon. Will always be in our memories."
Nicola Sweeney also paid tribute to Mr Lockhart and his family, saying: "Can't believe you are gone. God only takes the best."
Mr Lockhart's web page said he was a loft technician with an energy firm and that he studied law at the West of Scotland University.
Police Scotland's Sergeant Andrew Thomson, from Forth Valley Road Policing, said: "We are continuing our appeal for anyone who witnessed this tragic collision or who saw a black Fiat hatchback in the area prior to the incident at 2.20pm.
"We are particularly keen to trace the driver of a silver people carrier, which was seen in the area prior to the crash."
Anyone with information is asked to contact Police Scotland on 101 or Crimestoppers on 0800 555 111.