The visitors carved out numerous gilt-edged opportunities at White Hart Lane, but were continually denied by inspired goalkeeping from Hugo Lloris.
The France international twice kept out Olcay Sahan, while another close-range reaction stop from Demba Ba looked to have earned Spurs all three points after Harry Kane had put the home side ahead midway through the first half.
Besiktas ensured they left with a point, however, as Ba coolly tucked home an 89th-minute penalty following Vlad Chiriches' handball.
And Bilic, who spent time at both West Ham and Everton during his playing career, was torn between happiness and disappointment after the game.
"I am frustrated," he told ITV. "For every club it is always good when you get a point away at White Hart Lane but I think we played great football.
"We dominated, we passed the ball well and we definitely deserved more than this point.
"From our point of view it was a really good game of football but especially in the first half you have to score the kind of chances we had.
"To beat such a big club away you have to score those chances but of course I am proud of my team."
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
