The England international fired home to cancel out James Milner's first-half penalty and make it three draws in the last three meetings between the sides.
Jurgen Klopp had criticised his team's decision-making in last Saturday's shock 2-0 loss at Burnley and Philippe Coutinho twice spurned excellent chances to open the scoring in the first half, with Michel Vorm saving well.
Spurs struggled to threaten Simon Mignolet's goal for much of the first hour and Liverpool looked likely to double their lead, with Joel Matip hitting the crossbar and Sadio Mane's tap-in ruled out for offside.
But the visitors began to sit deeper as fatigue set in and Rose took advantage of some slack defending to drill home an equaliser preserve Spurs' unbeaten start to the campaign.
Mauricio Pochettino's side started well, but had Vorm to thank for a brilliant save to deny Coutinho with just four minutes played, the former Swansea City man sticking out a foot to deny the Brazilian who was unmarked just six yards from goal.
Christian Eriksen saw a free-kick beaten away by Mignolet and Dele Alli sliced a promising opening wide from 18 yards out as Spurs continued to look the more dangerous early on.
Liverpool, having begun to settle, applied some pressure of their own and it took two superbly timed sprints from goal from Vorm to deny Sadio Mane and Georginio Wijnaldum, before he saved another low shot from Coutinho from close range.
Kyle Walker was forced off through illness just before the half-hour mark, with Pochettino introducing Vincent Janssen in his stead in an effort to spark life back into Spurs' attacking play.
But it was down the hosts' right flank – now occupied by Eric Dier – that Liverpool found a breakthrough three minutes before half-time. Erik Lamela clipped the heels of Roberto Firmino after failing to stop his run into the area, and Milner swept the ball into Vorm's bottom-right corner from 12 yards.
Matip clipped the crossbar with a header eight minutes after the break, causing further unrest among the home fans, who were incensed when Mane – already booked – escaped punishment for a heavy challenge on Rose for the second time in the match.
Mane thought he had doubled Liverpool's lead in the 56th minute, but Adam Lallana was adjudged to have strayed offside before he squared the ball for the Senegal international to score.
Eriksen fired over from the edge of the area before Mignolet produced two fine saves to keep out Lamela's free-kick and Toby Alderweireld's header, as Spurs suddenly found some attacking impetus.
And they found their breakthrough in the 72nd minute to punish some poor Liverpool defending. Lamela flicked on Eriksen's cross from the right and Rose, arriving unmarked, had time to set himself and shoot low past Mignolet's near post to earn Spurs a share of the spoils.