Acting captain Mee was the man denied, both times from Brady corners, with his first header cleared off the line by Joel Matip and his second blocked by the legs of Mignolet at his near post.
Simon Mignolet then made a vital save to deny the Burnley captain. "They have to build; maybe Adam can start running this week or the beginning of next week", said Klopp, sounding optimistic. It would have been horribly unfair on a day when Liverpool had 35 shots and 72 per cent of the ball.
"Yes, result wise it is frustrating but honestly I am as a person not like this", said Klopp, who made seven changes from the Sevilla game. "I'm still in a 1-1 mood". September has been more sorry.
This will be Liverpool's first of three matches without Sadio Mané as he serves a ban following a red card against Manchester City. His side lost a lead against Sevilla in Europe.
Anfield is the next destination for Burnley in a unenviable start to the campaign that has already included trips to Chelsea and Tottenham, with Everton and Manchester City their next two league hosts. Liverpool went behind in the scoreline in the first half, caught up minutes later, and then spent the rest of the game beating their heads against a wall trying to get a victor.
They're heading to Anfield to take on Liverpool later.
Crane receives two four-shot penalties for non-conforming clubs
Had Crane noticed the the sticker before putting out on the 10th hole, the penalty would have only been two shots. Was penalized 8 shots for having tiny stickers on two clubs (help launch monitors collect data).
They had possession and pressure, chance after chance. From the first whistle, Burnley had looked to do just that, with Nick Pope taking an age on goal kicks and similar happening at throw-ins and every other opportunity. Shortly after Burnley's opener, Mohamed Salah did his thing; he took a long ball from Emre Can all the way from midfield, shimmied around two defenders, found a hole and punched through it. 1-1. He has started four games there.
However, the home side produced the flawless response as the impressive Salah controlled Emre Can's pass on the edge of the box before swiveling and finding the bottom corner with a low strike.
Burnley's stubborn resistance left a dominant Liverpool frustrated on Philippe Coutinho's return to the Reds' starting line-up. He had six efforts.
Dyche threw on Ashley Barnes - a scorer here last season to end the Clarets' 46-year wait for a league goal at Anfield - as an added outlet to try and reverse the flow.
It was all too avoidable.