The southbound service from Glasgow and Edinburgh to London had been due to arrive shortly after 7am, but did not reach Euston until around 10:30am.
Network Rail said there had been major signalling problems near Carstairs, between Glasgow and Edinburgh.
The locomotive hauling the train had to be replaced with another one in Edinburgh because of a suspected fault with electrical connections to the rest of the train.
The train arrived in Edinburgh at 1am, nearly two hours late, but did not leave until 2:48am - more than three-and-a-half hours late.
The delay is likely to have marred the trip for passengers late for appointments.
But some travellers will have enjoyed the chance to spend more time on the first of the £150 million fleet.
Carriages feature ensuite toilet and shower wet rooms, and double beds, in the service's biggest upgrade for 40 years.
Those aboard include Scotland Secretary David Mundell and Rupert Soames, chief executive of Serco, which runs the Sleeper.
Leading pollster Professor Sir John Curtice, of Strathclyde University, who was among other passengers, said: “It was a bit unfortunate, but I guess not surprising.
“Not helped by doing it in a night when there was a Network Rail-inducted diversion [via the east coast main line].
“I was travelling second class because it was so busy, so I will reserve my judgement on how much of an improvement it is - but at least the wi-fi made it possible to use the delay profitably.”
Network Rail said the signalling fault between Carstairs and Midcalder happened about 1140 - ten hours before the Sleeper left Glasgow.
It introduced "emergency special working" and repairs took until early this morning.
A spokesperson said: “A signalling fault caused significant delays to services on Sunday and we apologise to all those passengers who had their journeys disrupted.
"Our engineers were on site quickly to find the fault and begin repairs.
" However, the complex nature of the fault meant it took several hours before the repair works were competed."
Ryan Flaherty, Serco’s Managing Director for Caledonian Sleeper, said: “We apologise to guests affected by delays to our services last night.
"These were as a result of signalling problems across the network.
"We are engaging with Network Rail to understand why these issues occurred.”