Wilbraham denied the Rams a second league win this term when he finished from Bobby Reid's low cross.
Aden Flint's poor back header allowed Anya time to chip in the opener.
And the Rams will rue missed chances and the brilliance of former goalkeeper Frank Fielding for the failure to win.
Fielding made excellent second-half saves to deny James Wilson, Matej Vydra and substitute Darren Bent.
Derby were also on top before the interval, with Vydra shooting off target, Will Hughes sending a header just wide and Wilson also volleying wide.
The hosts rarely threatened with Abraham shooting over with their best chance.
Anya's cool finish finally ended Derby's miserable run of four league games without a goal and they looked set for maximum points despite their wastefulness.
However, Abraham and Reid combined to tee up Wilbraham who had a simple talk of scoring from a couple of yards out.
Bristol City manager Lee Johnson:  "I'm so pleased for Frank Fielding. Someone said well done in the dressing room afterwards and there was a spontaneous round of applause from the players because they all love him.
"He is so supportive to the other players even when things are not going right for him. He has had a difficult year recovering from injury but looked back to his best.
"Aaron Wilbraham is another player the whole squad respect. He polices the dressing room with his stature and remains an important member of the squad."
Derby County boss Nigel Pearson: "The reality is that it is a very big opportunity lost for us. We have created clear-cut chances that we have spurned at 1-0.
"When that happens it just takes a lucky break or a bit of good play with the opposition throwing extra men forward and we paid the penalty.
"You could see by the players' reaction at full-time how disappointed they were. We continue to stutter as a direct result of not converting the chances we are making.
"We have good players and to be fair to the lads we have had some very good performances over the last few games."
REACTION: Derby County manager Nigel Pearson speaks to BBC Radio Derby
Match ends, Bristol City 1, Derby County 1.
Second Half ends, Bristol City 1, Derby County 1.
Offside, Bristol City. Jamie Paterson tries a through ball, but Aaron Wilbraham is caught offside.
Goal!  Bristol City 1, Derby County 1. Aaron Wilbraham (Bristol City) right footed shot from very close range to the centre of the goal. Assisted by Bobby Reid.
Bobby Reid (Bristol City) wins a free kick in the defensive half.
Foul by Will Hughes (Derby County).
Attempt saved. Matej Vydra (Derby County) right footed shot from the centre of the box is saved in the centre of the goal. Assisted by Tom Ince.
Attempt missed. Aden Flint (Bristol City) right footed shot from outside the box is high and wide to the right. Assisted by Tammy Abraham with a headed pass.
Offside, Derby County. Andreas Weimann tries a through ball, but Matej Vydra is caught offside.
Foul by Hördur Bjorgvin Magnusson (Bristol City).
Matej Vydra (Derby County) wins a free kick on the left wing.
Attempt missed. Tammy Abraham (Bristol City) header from the centre of the box misses to the right. Assisted by Jamie Paterson with a cross.
Foul by Bobby Reid (Bristol City).
Scott Carson (Derby County) wins a free kick in the defensive half.
Scott Carson (Derby County) is shown the yellow card.
Attempt missed. Bobby Reid (Bristol City) left footed shot from outside the box is high and wide to the left. Assisted by Tammy Abraham with a headed pass.
Corner,  Derby County. Conceded by Hördur Bjorgvin Magnusson.
Attempt saved. Darren Bent (Derby County) left footed shot from the centre of the box is saved in the centre of the goal. Assisted by Matej Vydra.
Lee Tomlin (Bristol City) hits the bar with a right footed shot from outside the box.
Delay over. They are ready to continue.
Delay in match Bobby Reid (Bristol City) because of an injury.
Bobby Reid (Bristol City) wins a free kick on the right wing.
Foul by Craig Bryson (Derby County).
Corner,  Derby County. Conceded by Aden Flint.
Attempt missed. Aaron Wilbraham (Bristol City) header from the centre of the box misses to the right. Assisted by Jamie Paterson with a cross following a corner.
Corner,  Bristol City. Conceded by Darren Bent.
Attempt blocked. Lee Tomlin (Bristol City) right footed shot from outside the box is blocked.
Lee Tomlin (Bristol City) wins a free kick in the attacking half.
Foul by Will Hughes (Derby County).
Substitution, Derby County. Tom Ince replaces Jacob Butterfield.
Attempt missed. Craig Bryson (Derby County) left footed shot from the centre of the box misses to the right. Assisted by Matej Vydra with a headed pass.
Corner,  Derby County. Conceded by Luke Freeman.
Corner,  Bristol City. Conceded by Craig Bryson.
Substitution, Bristol City. Aaron Wilbraham replaces Marlon Pack.
Foul by Marlon Pack (Bristol City).
Will Hughes (Derby County) wins a free kick in the defensive half.
Attempt blocked. Lee Tomlin (Bristol City) right footed shot from outside the box is blocked.
Cyrus Christie (Derby County) is shown the yellow card for a bad foul.
Luke Freeman (Bristol City) wins a free kick in the attacking half.
Foul by Cyrus Christie (Derby County).
Aaron Wilbraham tapped home an injury-time equaliser for Bristol City after Ikechi Anya looked set to earn victory with Derby County's second Championship goal of the season.