Despite being down to 10 men from the 36th minute the hosts stood firm until the 73rd minute when James Husband broke the deadlock.
Shrewsbury thought they had snatched an unlikely point when goalkeeper Gary Woods somehow allowed a last-minute effort from Tom Eaves to squirm past him but Doncaster were not to be denied and Kyle Bennett earned them all three points.
"'We've had the benefit of seeing it on DVD and it's a scandalous sending-off," said Turner. "I've just been into see the referee who said he wasn't in control of his actions. What he's done is he's taken off to pass the ball or poke the ball to our full-back.
"The Doncaster player has been nowhere near the ball until a split second later and he's almost gone over David's backside so there was never any thoughts he wasn't in control and there was never any danger to the Doncaster player and we really do, on that basis, feel hard done to.
"But on the evidence of what we've seen we shall appeal against it."
Turner praised the character of his players who stuck to their task a man down but was just frustrated they emerged empty handed.
"It's hard enough to play against Doncaster who are top of the table now, but to go down to 10 men that early in the game made life very difficult for us," he added.
"Having said that we did show a bit of character. We worked hard and defended quite well and Chris Weale made one or two good saves."