Heineken Cup rugby was staged at the Sportsground for the first time and the French side ran in three tries as they dominated in Galway.
Connacht, backed by the biggest Sportsground crowd in their history of 9,120, found the going tough against the four-time European champions.
Elwood said: "I cannot fault the effort, we were up against quality.
"Toulouse were clinical, took their opportunities, and it was so, so difficult for us.
"To be honest we were under the cosh in the first half. Toulouse were business-like from when we found out what their team was yesterday and to be fair to them they were not taking any chances. They also had a good bench."
Leading 22-3 at half-time, with Lionel Beauxis kicking 17 points and Jean Bouilhou touching down, Toulouse could have expected to push on and grab the bonus point.
But they were denied by an immense Connacht effort in the second half, although Elwood said it was of little consolation.
His side's biggest reward was a penalty try after their pack shoved the French eight back towards their try-line.
He said: "I think we grew into the game, and we had a much better second half.
"But I am disappointed with the last score they got because at 29-10 we had a bit of momentum and we just overplayed the ball, and then they get another score which put a bad reflection on the scoreline.
"If they hadn't got that score (from Yannick Nyanga), we would have won the second half on the scoreboard."
Although Connacht enjoyed their share of possession, too many errors proved costly and Elwood acknowledged that the relentless pressure from Toulouse took its toll.
"Toulouse didn't miss a tackle, their breakdown work was immense, and that was the challenge.
"We knew we had to step our game up another level, we did our best, but they were sheer quality, in the physicality stakes, and in punishing you if you made a mistake.
"I am disappointed. Yes there was some good work today, some good individuals and passages of play, but I would like to have got into the game a bit earlier.
"Toulouse were chipping away and got a nice early 9-0 lead, so we were always playing catch up. We dug in there in the second half, we didn't completely fall off in the end, so there are a lot of positives and that's encouraging."
Toulouse forwards coach Yannick Bru was satisfied with the win, but disappointed his charges were denied a bonus point, particularly with Pool Six rivals Harlequins enjoying a good win over Gloucester.
"We are satisfied with the result because we knew it would be important to beat Connacht here, but the boys had a very good first half," he said.
"We said at half-time that we had our achieved first target and our second was to get the bonus point, but we didn't do it."
Bru said Toulouse had opted to start Vincent Clerc and Thierry Dusautoir to add strength to their side.
"We knew we had to be strong in mind coming here for Connacht's first home appearance in the Heineken Cup, so Clerc and Dusautoir had to be put on the pitch.
"We knew we had to be very good at the breakdown and contest every ball because Connacht have a fast game plan, and we had to stop the speed of the game.
"We did it in the first half and had many opportunities to score, but we are disappointed we had so many and finished with just three tries."
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