Bourgue, the French number 16, is ranked 164th in the world, but he led the British second seed two sets to one. He won 16 points in a row in an eight-game winning streak that helped him claim the second set in impressive fashion.
He lost 6-2 2-6 4-6 6-2 6-3 as he eventually ran out of gas.
"Losing against Murray is what I was expecting, so I wasn't really that tense," Bourgue told reporters.
"Then when I got the break, and then again, I all of a sudden was able to relax and sort of play with the crowd. I was able to manage and stay on top of my emotions.
"This makes me realise how much progress I have made over the past six, seven months."
Murray got off to a solid start, opening a 6-2 2-0 lead and there was no indication of the coming meltdown.
"I'm not sure I have ever had such a game level. It's interesting. I didn't feel like I was on the edge. I felt good," said Bourgue.
"Once again, he wasn't playing very rapidly. I felt he was giving me time to play, so I decided to just accept the rallies and every once in a while, you know, make a big shot. I did.
"At that point the overall game level was very high. I'm quite satisfied about some of the winning shots I was able to make."
Murray could only manage 12 points in the second set.
"There won't be many sets in the year where that's the case, so it's fairly rare," said the Scot.
"I was trying to find a way. It wasn't like I was not there mentally, but I just couldn't find the court. I was missing balls."
Next up for Murray is Croatian Ivo Karlovic.
(Reporting by Julien Pretot; Editing by Janet Lawrence)
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