Streller insisted United were "lucky" to salvage a point from the match after Ashley Young's last-minute header made it 3-3 at Old Trafford on Tuesday night.
United had been leading 2-0 at half-time, but Basel had missed a hatful of chances in the opening 45 minutes and it took a blast from coach Thorsten Fink at the break for the Swiss to make the most of their opportunities.
United boss Sir Alex Ferguson labelled his players "careless" and Streller said he was amazed by the ease with which Basel carved open the defence.
Streller told Press Association Sport: "I was really surprised at how many chances we got, as we were playing against one of the best teams in the world.
"But we have a coach who wants us to play attacking football and he said before that he wants us to go to Manchester aiming to win.
"Manchester did not have their best day and we had a really good day.
"The second half was one of the best halves I've ever seen from a Swiss team and in the end Man U were lucky we didn't win here.
"At half-time the coach was really angry and said it's not often you get so many chances against such an amazing team and to make sure we didn't miss any more in the second half.
"I'm very happy because you don't expect to get so many chances here.
"Now we really believe we can qualify from this group."