Several people at the Victoria Infirmary are said to have been left on trolleys on Monday night due to a "surge" in admissions.
Health board managers said a 24% increase in admissions at the hospital's A&E had led to "unacceptable" lengths of stay for some patients.
Other hospitals across the city also experienced a surge in admissions.
A total of 17 planned operations were cancelled at the Victoria Infirmary and 50 patients were found care home placements to free up beds.
The board also opened an additional nine in-patient beds in South Glasgow.
Emergency referrals from GPs were also diverted to other hospitals, although all urgent and cancer cases were unaffected.
Patients whose procedures were cancelled will have them re-booked as quickly as possible, the board said.
Robert Calderwood, NHS Greater Glasgow and Clyde chief executive, said: "I would like to apologise to those patients who experienced lengthy delays over the past 48 hours.
"The demands that we have been experiencing are unprecedented for this time of year and I would like
to pay tribute to my clinical colleagues in all our hospitals who have been working tirelessly to respond to this challenge."
He added: "We have put in place a number of steps to alleviate some of these pressures and have seen the situation improve.
"I also wish to apologise to those patients who have had the inconvenience of having their planned admission postponed.
"All patients affected will be re-booked quickly and given as early an alternative
admission date that suits."
The health board said health service had been a key part of the intensive multi-agency planning for the
Commonwealth Games and that it was "fully confident" it would be able to respond to "any eventuality".