The home side pulled one back through Dimitri Payet 10 minutes after the break but Leicester held firm at the back to see out their second straight win.
Bilic wanted his players to work harder on winning loose and long balls.
“We weren’t good enough at winning second balls and dealing with long balls," the 46-year-old told the club's website.
"Leicester were getting all of them, there are plenty of those situations in a football game, and if you lose the majority of them then it is impossible to win."
"You have to be reasonable, I can't be angry if they scored a good goal, and their second was a good goal. But I can't accept that too many times, we weren’t doing the basics, and that makes it much harder.
"We are not going to panic, we knew what we had before the game. For West Ham it is not that uncommon to beat Arsenal and then lose the next match," he added.
West Ham's surprise opening weekend win over Arsenal had come away from home and Bilic his players had to use their own home crowd as motivation, not a source of pressure.
"In football, you have many results like Saturday's where, somehow, clubs find it easier to play away games. They are not under the same pressure to win or to play well," the former Croatia manager said.
"On the other hand you have to use your home crowd to make it easier for you.
"Football is all about confidence; naturally you will have more confidence when you're 2-0 up then 2-0 down. Sometimes we have to take more responsibility with the ball."
West Ham host Premier League new boys Bournemouth on Saturday.
(Reporting by Shravanth Vijayakumar in Bengaluru; Editing by Peter Rutherford)
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