The Norway international admitted that the Cottagers would have taken a point before the game, but felt if they had been more clinical they could have snatched a win
A two-goal lead established by Olivier Giroud and Lukas Podolski saw Arsene Wenger's team cruising in the opening half hour, but a header from Dimitar Berbatov sparked a revival for the Cottagers that saw Alex Kacaniklic draw them level before the break.
A penalty from Berbatov in the second half briefly put the Cottagers ahead before Giroud levelled the scores again two minutes later.
Despite a late penalty save by Mark Schwarzer from Mikel Arteta, Riise believes the team could have come away with all three points had they built on their lead and been more clinical in their finishing.
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"But if you'd told us beforehand that we'd get a point, we would have taken it.
"Going 2-0 down, we thought it was going to be a hard afternoon. We did play well to start with but we conceded two silly goals. They then tried to control the game but we took over and scored two great goals of our own.
"We played well and caused a lot of problems and showed a great mentality and character to come back. At 2-2, I thought that we were going to win the game but then maybe it was fate to get a draw."
The 32-year-old commended his team's comeback and their ability to score freely this season, a feat that has only been bettered by league leaders Manchester United. However, the Norwegian has conceded that they need to improve defensively if they are to build upon their impressive start to the season.
"We're scoring quite a few, but we're conceding as well so that’s disappointing," Riise added. "It’s all about the points on the board, though, and we've got 16 so we should be happy.
"But we could have had quite a few more points. We’re going in the right direction, though."