The Hollywood-owned Welsh club's first point in the second tier of English football for 43 years was some consolation, but once again they were made to rue conceding late goals.
Two goals by Wales international Kieffer Moore, his first since a two-million-pound move, put Phil Parkinson's side in control, but Barry Bannan's reply meant a nervy finale.
The hosts failed to hold on with Bailey Cadamarteri levelling nine minutes from time.
Wrexham also conceded very late goals in a 2-1 home defeat by Southampton on the opening day of the season and again in a 3-2 home defeat by West Bromwich Albion.
An unprecedented three successive promotions have meant Wrexham, bought by Hollywood actors Ryan Reynolds and Rob McElhenney in 2021, began the season with high hopes of making it four and reaching the Premier League.
But the early indications are that they may find it tough to live up to the sky-high expectations.
After three games, they are 19th of the 24 teams.