With the vital away goal in the bag, it will leave the English Premier League side with an excellent chance of progressing in a competition which offers a Champions League place to the winner.
Klopp spent seven successful seasons at Dortmund and he received the warmest of receptions in the Westfalenstadion, the 65,000 crowd with a rendition of "You'll Never Walk Alone" -- the signature song for visiting Liverpool supporters.
Once the action was underway, Klopp's new charges did him proud with Divock Origi striking to put the visitor ahead in the 36th minute.
His low shot after wriggling free of defenders found the net after an assist from Jordan Henderson, who did not come out for the second half due to injury.
Origi might have increased the lead just before half time, but soon after the break Dortmund was level.
From a short corner, Henrikh Mkhitaryan whipped in his cross and Mats Hummels powered home the header, his first goal in European football since 2013.
Both teams had chances to secure the winner, with Dortmund keeper Roman Weidenfeller denying Philippe Coutinho with a brilliant save and keeping out two further efforts in quick succession.
By the finish, the 4,000 contingent of away fans was making the most noise and it was a contented Klopp at the finish.
"I'm pretty sure a lot of people thought we'd lose 2, 3, 4-0. We had them in some moments. Around our goal we were brilliant," he told BT Sport.
Klopp's return has been headline news in Germany all week, while on social media it earned the hashtag #ElKloppico -- a reference to the famous El Clasico clash in Spain between Real Madrid and Barcelona.
In other ties Thursday, visitor Sevilla edged Athletic Bilbao 2-1 in an all-Spanish clash, Shakhtar Donetsk won 2-1 in Braga and Villarreal secured a 2-1 home win over Sparta Prague.
The second legs will be played next Thursday.