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, who scored through Imanol Harinordoquy's late try while Dimitri Yachvili kicked three penalties.
"It is a huge scalp for us. Biarritz had a full squad out there and we had a lot of guys missing," Elwood 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 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."
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 the two-time finalists 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 and were beaten by a better team."
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."