Dean Ryan's men closed the score to 25-15 and heaped pressure on Leicester as they looked to secure a losing bonus-point.
Leicester eventually made the victory look more comfortable courtesy of a last-minute Vereniki Goneva try but, according to Matavesi, his side should be pleased with their performance.
He said: "It was always going to be tough coming to Welford Road for the first game of the season. But the boys did well and made a good game of it.
"It was a tricky first 40 minutes. It was pretty much the first time we'd played together as a team in a competitive match.
"So it was really pleasing that we got our heads together in the second-half, played a bit of rugby and put Leicester under pressure."
Matavesi produced a sensational solo try as he burst his way through and looked dangerous throughout as he ran at the Leicester backline.
And he believes his side can pick-up some good results this season if they can keep pressure on teams for long periods.
He continued: "I just saw the gap and went for it, so it was great to get the try. We just need to be a bit more consistent rather than playing in patches.
"We need to put teams under pressure for 80 minutes. If we don't, teams like Leicester will put points on you and it is hard to claw them back."
Warriors return to training tomorrow as they switch their attention to next Saturday's clash with London Irish. And Matavesi has urged fans to get behind the side when Warriors open their Sixways season.
He added: "It's a quick turnaround. We'll get our recovery done tomorrow and look at the Leicester game.
"But then we'll move on to London Irish. It's a new season at home so hopefully we'll get the crowd behind us.
"We'll target a win and it will be great to play in front of our own fans again. We're exciting to watch when we get things right and we showed how dangerous we can be today. So there's lots for us to build on."