Shrewsbury have been widely tipped to struggle by the bookmakers this season but they created the better chances against the Dons.
And had it not been for a series of fine second-half stops from visiting keeper David Martin, who twice denied John Marsden before somehow blocking a close-range effort from Aaron Wildig, Turner would have been even happier.
"The encouraging part about it is we can get better and that's the pleasing thing," said the vastly-experienced Turner. "It's always difficult on the first game.
"As a manager, you're never quite sure what your best side is, whether you've worked hard enough in pre-season, had enough games, whatever it is.
"It's not until you get two or three games into the season that you start to get a measure of yourselves.
"This was a good indication that we look fit, we looked strong, but we just needed a little bit more here and there."
Turner was pleased to see his team stand firm to register a clean sheet with new captain Tamika Mkandawire impressing while he also picked out the contribution of consistent left-back Joe Jacobson.
Turner added: "I thought we defended in the main quite well. There was one shot Chris Weale tipped over. I think it was going over anyway but it was a good stop.
"It's always good for defenders, good for a team, to keep a clean sheet, and we've worked hard as a team to achieve that. We just need a little bit more going forward now.
"I thought Tam played extremely well and Joe Jacobson had a superb game. In the second half he was dominant and he got into some great positions because we were in trouble a couple of times and his defending at the back post was absolutely first class."
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