Goals in either half from Michail Antonio and Jesse Lingard, and a resolute defensive display, earned a rare derby win, though Spurs dominated possession throughout.
"It's good, but I've told them we've only done half a job," Moyes told Sky Sports.
"If we continue this way we will be close, so I don't want to be negative in any way, it's a great result for us to beat Spurs.
"With the ambitions they have got and what they have at their disposal, so we're enjoying it, but we're not going to be daft, we'd be stupid to say anything.
"We're going to keep calm about it and hopefully keep pushing and pressurising the teams near the top."
Spurs face both Manchester City and United, plus Leeds and Arsenal before April in a run on season-defining fixtures.