Chelsea had two goals in the first half disallowed before Lukaku opened the scoring with a second-half penalty to bag his first goal for the club in 2022. The Belgium international then doubled their lead just two minutes later with a 20-yard strike.
However, Francisco Trincao scored for Wolves with 10 minutes to go before Coady netted in the 97th minute to grab Wolves a point.
The game was watched by Los Angeles Dodgers part-owner Todd Boehly, whose consortium won the bid to acquire ownership of Chelsea in the early hours of Saturday.
Chelsea dominated possession in the early stages, but failed to take advantage of their control with Timo Werner, Romelu Lukaku and Christian Pulisic all going close before Wolves found a foot in the game with a couple of chances of their own from Romain Saiss and Ruben Neves.
The deadlock looked to be broken when Timo Werner jumped on a beautifully weighted Pulisic ball on 34 minutes and nudged the ball past Wolves goalkeeper Jose Sa, but the goal was disallowed for a foul on Saiss in the build-up.
Chelsea had a second goal disallowed just five minutes later with Ruben Loftus-Cheek deemed offside when he poked in the ball from a corner, sparking protests from the Stamford Bridge crowd.
The hosts started the second half brightly and were awarded a penalty on 55 minutes when Lukaku was caught by Saiss' trailing leg in the box. He sent Sa the wrong way to convert the sport kick and put Chelsea ahead with his first goal for the club since December.
Lukaku doubled Chelsea's lead just two minutes later with a powerful 20-yard strike after Pulisic stole possession from the Wolves defence.
Substitute Trincao did find a breakthrough on 79 minutes, sending a left-footed shot past Mendy from the edge of the box.
It all looked to be over for the visitors, but with the last shot of the game, Coady found the back of the net from a header, powering in a Chiquinho cross from six yards out to secure the draw.