"Rory (Best) said there were guys just hanging on in there after 50 minutes.
"I feel very sorry for the guys because they were just exhausted but they played really positive, good rugby and while you could questions a few of their decisions on the field you could never question their courage and commitment.
"They pushed themselves right to the very end, tried their guts out and just couldn't ask anymore."
In their first home defeat since Williams took over, Ulster trailed by 15-12 at the break but they muscled their way into a 24-22 lead thanks to tries from wing duo Simon Danielli and Tommy Bowe.
However, Glasgow regained control in the closing quarter-hour against a tiring Ulster side.
"In all honesty I thought we were the better side and the ball-in-hand stuff we played was superb and we really went for it but we let in too many soft tries," the Australian said.
"We conceded four tries - two in each half - with no tackles on the ball carrier and you can't do that and from there Parksie controlled the game."
Williams and his players did have reason to celebrate on Saturday afternoon as Connacht's 39-11 defeat to Cardiff the previous night guaranteed Ulster's presence in next season's Heineken Cup.
"I'm disappointed we lost at home because we wanted to celebrate getting into Europe," Williams added.
"But I hope that in the future the officials take note that if we miss a game that it is made up in a spare weekend because we had enough spare weekends to do that. This was a situation that should never have come up."