Murray and Dan Evans saved two match points to defeat the Belgium duo Sander Gille and Joran Vliegen 6-3 6-7 (10-8) 11-9 and reach the quarter-finals.
It was another great escape for the pair, who saved five match points in the previous round, and their heroics left Murray in tears as he slumped back in his chair.
Afterwards, he said: "It's unbelievably emotional. I mean you're obviously unbelievably happy and then for whatever reason, I don't know why it happens, it's like happy tears.
"I don't know why that's the case but I was really emotional at the end of the match.
"Obviously, I was extremely happy, excited that we managed to get through."
The three-time grand slam champion and one-time Olympic gold medallist is still hopeful of going further in the competition, having fallen out at the quarter-final stage of Tokyo 2021 alongside Joe Salisbury.
Murray said: "The way the scoring system is, and the way the teams play now, the margins are just so fine that anything can happen really.
"But we've definitely got a chance of getting through to the semis."
Evans, who is set to lose more than 110 ranking spots for not defending his Washington Open title in order to team up with Murray in Paris, is pleased with the improved level of performances as the tournament has progressed.
He said: "What happened again was incredible.
"We're getting really close to doing something pretty special.
"People see that, how good we've been in tight moments, and we're getting better and better."
Murray and Evans will be back on court on Thursday and are now just one match away from the chance of a medal for Team GB.
Murray's glittering career came to an end on Thursday in the Olympics men’s double quarter-finals as he and Dan Evans were beaten in straight sets by American pair Tommy Paul and Taylor Fritz.
He announced before the Games that it would be the final tournament of his career, and he pulled off two superb comeback wins, saving multiple match points in both, to make it to the last eight.
Murray bows out as a two-time Olympic gold medallist with three grand slams. He won 739 games across his 1001 ATP Tour level matches, winning 46 singles titles.
Five of those came at Queen's Club, and shortly after his retirement was confirmed, the newly named Andy Murray Arena was announced.
"Andy is the greatest tennis player ever to come from this country and a giant of British sport," LTA chief executive Scott Lloyd said in a statement.
"His contribution to the game is immense and has bought us all so many moments of pride. He was relentless in his pursuit of excellence and had a single-minded determination to succeed.
"We are proud to name the arena at our tournament at the Queen's Club after him."
Meanwhile, Novak Djokovic has led the tributes to Murray, having played him 36 times over the course of their careers.
"Andy, we were all hoping that this message will never come, but here it is," Djokovic said in a message uploaded to the ATP Tour's X account.
"I just want to say congratulations for an amazing career, multiple grand slams, number one in the world, two Olympic gold medals, a Davis Cup. You've done it all. Amazing, really.
"Congratulations to you, to your family. We go back a long time. I think we were 11 when we met the first time on the court, and we had some incredible encounters, and rivalries.
"We played finals of every single grand slam. You brought the Wimbledon title to Britain after 77 years, unfortunately, against me in the final, but it was quite a special historic moment that I cherish and was grateful to be part of.
"You can be proud of your incredible resilience and fighting spirit, the inspiration that you give to all of us after all the hardships you went through - the surgery, the restrictions in your movement.
"You still kept going, you showed everyone there is no letting go and no giving up. Kudos to you and all the best to you and your family. You deserve the best possible farewell."
There had long been speculation around how long Murray would continue to compete, with his career beset by injury issues that limited him to three major titles.
But heading to France this week, the two-time Olympic champion confirmed the Games would be his final tournament.
In an interview with BBC Sport, Murray revealed only now is he entirely content with a decision that had long seemed likely.
"I didn't feel that way a few months ago when I thought that this is when I was going to stop," Murray said.
"I didn't want to. Now I want to. I know it is the right time for me."
Murray is still managing a back injury, but he is in line to compete in both the singles and doubles events, teaming up with Dan Evans in pursuit of another medal.
"Me and Dan made the commitment to each other, that is what we will prioritise," he added.
"That gives the team and us the best opportunity to get a medal.
"My back is still not perfect. I didn't feel great in the build-up to Wimbledon and during it. I feel better here."
