Sunderland, who battled hard to cancel out Sergio Aguero's 16th-minute strike, are mired in 19th place with 19 points from 24 games.
"We missed too many chances and when you play a Manchester City side and have better chances than them, it shows the level of our performance," Allardyce told reporters.
"There's no result for us, the points aren't there, but the positives are. While this performance was the best since I've been here, we're in big trouble now when you look at the other results."
Sunderland play away at Liverpool on Saturday.
(Reporting by Shravanth Vijayakumar in Bengaluru; editing by Justin Palmer)
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