The Brazilian-born striker, who joined Shakhtar from Arsenal in the close season, was mobbed by his team mates after striking just before the interval but kept his head down while home fans ignored near freezing conditions to celebrate bare-chested.
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"It is a missed opportunity of course," Arsenal manager Arsene Wenger told Sky Sports.
"The hardest thing to do in a big game is score the first goal and that is what we did but then I think it's a lack of maturity not to focus 100 percent and get caught just before half-time."
Speedy striker Theo Walcott stunned the Ukrainian champions 10 minutes into the game when he raced through Shakhtar's half and caught keeper Andriy Pyatov napping with a low strike into the far corner.
The home side equalised 18 minutes later when Dmytro Chygrynskiy's header was slightly deflected by Arsenal defender Craig Eastmond into his own net and Eduardo's strike brought a near capacity 50,000-strong home crowd, including Ukrainian President Viktor Yanukovich, to their feet.
"We thought we could get through without 100 percent focus on every defensive situation," said Wenger.
"In the second half we gave a lot to come back into the game and were a bit unlucky with the finishing but they were intelligent as well and that explains why we couldn't come back. We had the chances but we could not take them."
Shakhtar's Brazilian midfielder Jadson twice wasted great opportunities shortly after Walcott's strike, first hitting a defender with the goal at his mercy then, just a few seconds later, heading against the post.
Pyatov then made several fine saves, first denying Sebastien Squillaci with a point-blank header, then stopping Tomas Rosicky's low drive from inside the box.
Substitute Carlos Vela missed Arsenal's last big chance with five minutes remaining when he fired straight at Pyat
