A superb finish from Liverpool striker Divock Origi was cancelled out by a bullet header from Dortmund captain Mats Hummels.
The result favours the English side ahead of the return at Anfield in a week’s time.
“Now there’s a real bit of spice in the tie, but we haven’t got the feeling we’re through,” said Klopp.
“Things could have gone worse and all in all it was good a good result from our perspective. Around the time we went 1-0 up, we really showed what we can do.”
On his much-hyped return to the Ruhr Valley, Klopp received a warm welcome at Dortmund’s Signal Iduna Park, where he enjoyed seven years as Borussia coach, winning two Bundesliga titles and the German Cup in 2012.
His name received a loud cheer from the home fans when it was read out before kick-off and he hugged Dortmund CEO Hans-Joachim Watzke on the halfway line during the warm-up.
“It was nice to be back and I really valued the applause I had before kick off,” added Klopp.
Both sets of supporters gave a rousing rendition of the two clubs’ anthem ‘You’ll Never Walk Alone’ in front of a sell-out crowd of 65,848, but once kick-off came it was down to business.
Dortmund made the stronger start with two clear chances in the opening half.
With a quarter of an hour gone, Marco Reus snapped up Adam Lallana’s clearance but fired his low drive just wide of Liverpool goalkeeper Simon Mignolet’s right post.
The game had to be paused in the 21st minute when James Milner’s free-kick found Dejan Lovren. The Croatian clashed heads with Roman Weidenfeller in the goal-mouth scramble and the Dortmund goalkeeper needed medical treatment before he could carry on.
Dortmund then wasted a golden chance when the Liverpool defence left Henrikh Mkhitaryan unmarked, but the midfielder dragged his shot just wide in the 34th minute.
And the hosts were left to regret their missed opportunities when Liverpool struck.
Milner, who was superb all night, flicked a pass to Origi and the Belgian striker slotted past Weidenfeller with a superb low drive, and a slight deflection, with 36 minutes gone.
Klopp produced his trademark pumped-fist celebration on the touchline and Liverpool almost got a second when Milner’s shot was blocked by Lukasz Piszczek in the 40th minute.
Dortmund coach Thomas Tuchel brought on ex-Liverpool midfielder Nuri Sahin for Germany international Erik Durm at the start of the second half.
It gave the hosts fresh impetus, and they were level within three minutes of the restart.
Hummels attacked a Mkhitaryan cross and got up above Lallana to head home the equaliser.
However, only a brilliant save from Weidenfeller just moments later stopped Liverpool going 2-1 up.
Philippe Coutinho found space and hit his shot towards the bottom left corner, but the Dortmund shot-stopper got down in time as it finished all square.
“We gave the goal away too easily, but all in all, the point will do, even if we had hoped for more,” said Dortmund’s goal-scoring captain Hummels.
Dortmund were arguably better on the night, but Klopp’s Liverpool will start next Thursday’s return leg at Anfield as favourites to reach the semi-finals.
“We came here to get the win and we did well today,” said Liverpool’s Germany defender Emre Can.
“A 1-1 draw is a good result, but the chance of going through is still only 50-50.”