Dan Parks delighted the Sportsground faithful with a 17-point kicking contribution, Tongan winger Fetu'u Vainikolo swooped for a first-half try and a Mike McCarthy-inspired pack delivered the goods against their much-vaunted opponents.
It was only Connacht's third victory in Heineken Cup history, adding to previous successes against Harlequins and Zebre, and leaves them second behind Quins in the Pool Three standings.
Elwood, in his final season in charge of the Irish province, said his initial feeling was one of "sheer delight and joy".
"It is a huge scalp for us. Biarritz had a full squad out there tonight and we had a lot of guys missing," he said, referring to a Connacht injury list that includes Michael Swift, regular captain Gavin Duffy, John Muldoon, Nathan White, Frank Murphy and George Naoupu.
"A lot of credit must go our young guys, particularly in the forward pack. The platform from our forward pack was terrific.
"To get scrum penalties against Biarritz was magnificent and Dan (Parks) kept the scoreboard ticking over, so all in all it was great."
Elwood was thankful Biarritz got nothing out of the game, despite them grabbing a try at the death through replacement Imanol Harinordoquy.
"Thankfully it finished in our favour and they got nothing out of the game," he said. "That is a good feeling, it would have been awful if they had got a bonus point [with the conversion] at the end."
Elwood said it was important to win European games at home, but he expects a tough visit to the Basque country in the return fixture next weekend.
"Now we are under no illusion that there will be a backlash, but this win sets us up nicely and we are going to France with a lot confidence," he said.
Biarritz were sent home from Galway with their tails between their legs, according to full-back Iain Balshaw.
The shock defeat will make qualification for two-time finalists Biarritz difficult, though still possible.
"It is bitterly disappointing," said Balshaw. "Mathematically there is a chance we can quality, but it will be very tough. Presumably we will need to get five points from every match here in.
"It is going to be tough. There was not much said in the changing room afterwards, but a lot will be said on Monday. We were not good enough tonight and were beaten by a better team."
The former England international said Connacht were better at the set-piece, breakdown and at ball retention.
"I don't know how many turnovers we made, probably three times as many as Connacht and at this level you cannot win rugby matches [playing like that], not just in the Heineken Cup but in the Top 14," he said. "Unfortunately, we just do not have the consistency at the moment.
"I think there are a lot of individuals who have to work on their own game. It is something that is not just happening this season, but last season as well and the individual error count is killing us."
Balshaw expects the Biarritz supporters to be "annoyed", adding: "We are disappointed because we focus on the Heineken Cup. We want to make the knockout stages, but if you perform like that you will never make the knockout stages. We have got to improve drastically."
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