Bergwijn, who enjoyed an evening to forget in front of goal, also missed a glorious chance within seconds of the restart, rolling wide of the far post, while Kane forced Alisson into a one-handed stop.
The misses left Spurs wondering what could have been as Liverpool maintained their remarkable record at Anfield and underlined why they are favourites to retain the title, and Spurs still rank outsiders.
In the first half, Mohamed Salah had given the hosts the lead with a deflected effort but Spurs hit back through Heung-min Son, who rolled home after beating the offside trap.
Liverpool dominated a contest that predictably proved a clash of styles, with Mourinho’s side lying in wait for a Liverpool mistake in their own half while the Reds enjoyed almost total control of possession as they probed for an opening.
Mourinho may ultimately reflect that Spurs’ game-plan worked, as they frustrated Liverpool but missed their big opportunities to record a landmark victory.
The mid-week results elsewhere could scarcely have gone better for both sides, as Man City, Chelsea, Southampton and Leicester all dropped points, relieving some of the pressure at Anfield, particularly on Spurs.
Free-hit or not, it was, however, inconceivable that Mourinho would open up against the champions and Spurs were set-up to contain-and-counter again, with Moussa Sissoko on the right of a midfield four to shackle Robertson.
Spurs’ first tactical call appeared to be a swap of ends following the coin-toss, so the hosts played towards the majority of their 2,000 supporters in the Kop – who paid an emotional tribute to former Reds boss Gerard Houllier before kick-off – in the first half.
There was an element of luck about Liverpool’s opener, with Salah’s effort taking a huge deflection off Toby Alderweireld and looping home off the post.
Spurs had enjoyed little more than the occasional glimpse, usually when Kane had managed to get on the ball, but their deep-lying approach paid off with an equaliser within seven minutes.
Lo Celso justified his selection with a fine outside-of-the-boot pass to release Son, who raced clear to scored a trademark goal under Alisson.
It was exactly the type of goal Mourinho’s side is set-up to score and took Son to 11 in the League, almost all of which have come from racing behind high defensive lines. The only surprise was that Kane did not assist it.
Within four minutes of the second half, Spurs had more good chances than in the whole of the first period. Bergwijn fired wide within seconds of the restart when he really should have scored and Kane forced Alisson into a save after intercepting the goalkeeper’s pass.
Then came the minute that Spurs will look back on in agony. First, Bergwijn got behind Liverpool’s defence but could only strike the base of the post.
Then, from substitute Lucas Moura’s corner, Kane headed into the ground and over the bar when it was easier to hit the target.
Liverpool so often have made teams pay for profligacy and so it proved. Spurs dropped deeper and deeper in search of a point but Firmino rose above Kane and Alderweireld at the death to illustrate the difference between champions and pretenders.
Spurs must find a way to accommodate Lo Celso and Ndombele
Mourinho promised before the match that eventually Lo Celso and Ndombele would play together in Tottenham’s midfield. That moment cannot come soon enough if Spurs want to be keep the pace at the top of the table.
Lo Celso justified his selection, yet Spurs missed Ndombele’s ability to keep hold of the possession as they squandered the ball time and again, increasing the pressure which eventually led to Firmino’s sucker-punch.
Mourinho is unlikely to drop Sissoko, who offers physicality, energy and tactical discipline, but perhaps Bergwijn could be the fall guy.
Losing the Dutchman, though, who deprive Spurs of a runner in behind, which is so crucial in this type of match. On another day he might’ve scored twice and been the hero.
It is a difficult conundrum for Mourinho but one he must solve to take his side to another level.
Tottenham’s profligacy at Anfield raised a serious question beyond the significance of the result: have Spurs got enough goals in the team?
Beyond Kane and Son, who have scored 20 between them, no Spurs player has scored more than once this season and Bergwijn, who is most often their third forward in the League, showed his limitations with two big misses.
Spurs look particularly short of goals in midfield, where Pierre-Emile Hojbjerg and Sissoko are guaranteed starters but never likely to score. Lo Celso and Ndombele are both only occasional threats in the final third.
Gareth Bale, missing through illness again, would improve the situation if he gets fit and firing and so too would Dele Alli, who made another late cameo, suggesting he is slowly being reintegrated to the side.
But for now, Spurs are massively reliant on their superstar front two, which should worry Mourinho after a wasteful and frustrating evening.