Emre Can cancelled out Marko Devic's well-taken opener for Rubin to notch the first goal of the Klopp era, but despite playing against 10 men for 54 minutes following Oleg Kuzmin's dismissal, Liverpool could not find a winner against a side who are currently 12th in the Russian Premier League.
The fit-again Christian Benteke, on as a second-half substitute, came closest with a shot that hit the post, but he could not prevent Liverpool registering a third successive 1-1 draw in Group B and the second stalemate of Klopp's tenure after Saturday's 0-0 draw at Tottenham Hotspur.
"When you come for the first time to a new house, you normally get a present," Klopp said.
"I'm not satisfied with my present tonight, but it's only the first time and I'll come again. We'll see next time. Maybe it's better."
The one consolation was that Liverpool held on to second place in the group ahead of next month's return fixture in Kazan, although they are now four points below leaders FC Sion.
Klopp's arrival was excitedly greeted outside Anfield, with scarves and hats bearing his image adorning stalls and Klopp-branded T-shirts, mugs and pennants available in the club shop, but his team's performance demonstrated the work required to make Liverpool a force again.
He had swapped the tracksuit he wore at White Hart Lane for a black three-piece suit and he was given an enthusiastic reception as he walked out to survey the warm-ups prior to kick-off.
And yet in a reflection of Liverpool's recent problems, as well as a low-key Europa League campaign that had previously yielded 1-1 draws against Bordeaux and Sion, the atmosphere was subdued.
Liverpool looked to enliven the crowd in the early stages, with Can and Nathaniel Clyne shooting off-target.
But from Rubin's first attack, they scored. Devic crept in behind Clyne to chest down Kuzmin's lofted pass from the right and brilliantly beat Simon Mignolet with the outside of his right foot.
- Firmino returns -
Liverpool resumed their assault on the visitors' goal, with Adam Lallana and Philippe Coutinho each going close on a couple of occasions, but Blagoy Georgiev served a reminder of Rubin's threat on the break with a shot that Mignolet athletically palmed over the crossbar.
Liverpool needed a spark and nine minutes before half-time they received a big helping hand when Kuzmin was shown a second yellow card for impeding Can.
It was a stroke of good fortune for the hosts and they took immediate advantage, with Divock Origi nodding down Coutinho's deep free-kick for Can to crash home from close range.
Watching on from the edge of his technical area, Klopp jutted out both fists in celebration.
He might have had another goal to cheer before half-time, but after shimmying into space on the edge of the box, Coutinho fired wide.
With Rubin a man down, the Russians struggled to get out of their own half in the second period.
Liverpool had no trouble getting to the edge of their opponents' box, but it was there that things went awry, with James Milner, Clyne, Coutinho and Can all failing to extend goalkeeper Sergei Ryzhikov.
As the game entered its final third, Klopp threw on Benteke, back after five games out nursing a hamstring injury, and then Roberto Firmino, who has overcome a back problem. Coutinho and Origi made way.
Benteke could not solve Liverpool's profligacy, however, first volleying over when unmarked from Clyne's deep cross and then placing a low shot against the right-hand post.
On spoiling Klopp's big night, visiting coach Valery Chaly developed a theme that the German had used himself.
"We're not relatives here to give him some presents," he said. "This is football. Both teams gave a decent performance and we're happy with the result."