McInnes watched on in frustration as Thomson failed to give a spot-kick when Niall McGinn's flicked shot struck Dundee defender Willie Dyer on the arm four minutes before the interval.
But a lack of guile also proved costly for Aberdeen, who are now five points off the pace with Celtic having played one game fewer.
McInnes said: "It was such a pivotal moment. It's a blatant penalty. It has cost us today because they've managed to get an equaliser after that.
"It's not all down to the referee, though, that we haven't won the game as we're still in a good position at 1-0. We had an excellent start and Dundee were forced into playing like an away team.
"I thought we were in charge of the game but gave Dundee encouragement by being a bit ponderous in the second half.
"We had umpteen set-plays but we've been here before. In the second half, we let Dundee back into the game and the pitch didn't help.
"But from the minute Dundee got their equaliser I thought they were worthy of the point to be fair."
Source: Teamtalk