Liverpool also suffered another injury concern when Joel Matip limped off with a groin injury in the second half.
West Brom earned their first point under new boss Sam Allardyce but remain in the relegation zone.
It also snapped Liverpool's seven-game winning streak at home as Allardyce, beaten 3-0 by Aston Villa on his debut last Sunday, reinvigorated West Brom's hopes of avoiding relegation from the top flight.
Klopp vented his frustration with West Brom's defensive approach while Allardyce was purring with delight.
"It is an incredible challenge when you constantly face 10 men and I don't think anyone can expect a team that plays like West Brom in the first half, like a 6-4 or whatever it was," Klopp told Sky Sports.
"To be fair West Brom did their job inside 90 minutes. West Brom deserved the point tonight because we didn't finish the game. It is difficult to create. In the end everyone knows and feels we should have and could have done better tonight."
The Baggies still have a mountain to climb if they are to avoid relegation as they sit 19th on eight points from 15 games, five adrift of safety with 17th-placed Burnley also having a game in hand.
"It was superb today and we came out with a precious point for us," Allardyce, who has managed 11 clubs without ever getting relegated, told the BBC.
"There is a lot of defending you have to do against a quality team with a fantastic home record. To the man, we frustrated Liverpool as much as we could. We had to take the sting out of them in the first half.
Liverpool dominated possession from the first minute and opened the scoring after 12 minutes when Matip clipped the ball into Mane who chested it down and rifled a shot past West Brom goalkeeper Sam Johnstone.
By half-time, Liverpool had enjoyed 87% possession and no West Brom player had completed more than eight passes.
The visitors provided a better threat in the second half and were given a boost after Matip limped off after 59 minutes.
Liverpool continued to be frustrated and West Brom almost grabbed an equaliser when Karlan Grant was put through but Alisson blocked his shot
West Brom grabbed their equaliser after 82 minutes when Ajayi headed in from Matheus Pereira's cross.
Roberto Firmino missed a 90th-minute chance to snatch a win for Liverpool as Johnstone somehow clawed out his downward close-range header and turned it around the post.
"When you go slack you get what you deserve," Robertson told Sky Sports. "When you come up against a Sam Allardyce team you know they will stay in it and take the one chance they get.
"We created chances in the first half and then in the second half it was as if we didn't want to do it again. Every one of us made a mistake in the second half which can't happen.
"We will take the point and move on. We can't dwell on it but it's a huge disappointment."