The Scot was two sets to one up overnight when the 11pm curfew came into play but he was unable to complete the job, with fifth seed Tsitsipas fighting back to win 7-6 (7-3) 6-7 (2-7) 4-6 7-6 (7-3) 6-4 on Friday.
It was a hugely disappointing way for Murray to mark the 10th anniversary of his career-defining first Wimbledon title, and he is all too aware that his chances for another deep run here are ebbing away.
The match began under the roof on Thursday but there were blue skies above on the hottest day of the tournament so far as Murray returned two sets to one up.
Murray had clearly been eager to avoid the lottery of another tie-break and, in a repeat of the first-set shoot-out, it was Tsitsipas who won the final four points.
Murray's strategy to relentlessly probe the Greek's backhand was perhaps becoming a little predictable, and he was in serious trouble when Tsitsipas created three break points in the third game of the deciding set, taking the third to break serve for the first time.
The home favourite saved two match points but Tsitsipas clinched it on his third chance with his 17th ace to book a third-round clash with Laslo Djere.
There was more disappointment for local fans when last year's semi-finalist Cameron Norrie also got knocked out, beaten 6-3 3-6 6-2 7-6 (7-3) by American Christopher Eubanks.
Liam Broady was the only British man to make the third round but his singles campaign ended 90 minutes after Norrie's with a 4-6 6-2 7-5 7-5 loss to Denis Shapovalov.
He took no comfort from going the furthest, saying: "It kind of makes it worse that we've all lost on the same day.
"Obviously I'd have preferred for us all to have won. Probably would have felt a bit better if we'd staggered what days we lost on, whereas now it's a bit of a Debbie Downer, isn't it?"
World No.1 Carlos Alcaraz had earlier had to work hard for his first Centre Court win, earning a 6-4 7-6 (7-2) 6-3 second-round victory over Alexandre Muller.
It made it seven consecutive wins on grass for the top seed, who is viewed by many as the most likely candidate to stop defending champion Novak Djokovic in his pursuit of a record-equalling eighth title at the All England Club.
"Of course, it's really special ... A really beautiful court," said Alcaraz. "I feel that I belong in that court. It would be amazing for me play a final here in Wimbledon. Even better if is against Novak, obviously."
Alcaraz made 41 unforced errors and did not have it all his own way against an obstinate opponent, who clearly enjoyed his first outing on the big stage.
No.3 seed Daniil Medvedev took a while to get up to speed in his first-round match against British wildcard Arthur Fery but was impressive in a 6-3 6-3 7-6 (7-5) win over the dangerous Frenchman Adrian Mannarino.
Olympic champion Alexander Zverev defeated Yosuke Watanuki 6-4 5-7 6-2 6-2 and will face Matteo Berrettini, who hammered Alex de Minaur in straight sets.
Last year's quarter-finalist Jannik Sinner recovered from a sluggish start to beat unseeded Frenchman Quentin Halys 3-6 6-2 6-3 6-4.
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