The Belgium striker struck on Reds boss Jurgen Klopp's return to his former club with a low finish across goal.
Dortmund goalkeeper Roman Weidenfeller then denied Origi one-on-one.
Mats Hummels headed in an equaliser and the hosts pressed in search of a winner, but Liverpool kept them at bay.
In the end it was a fair result as Dortmund, favourites to win the Europa League, dominated possession but Klopp's tactics ensured Liverpool frustrated their opponents, restricting them to few real chances.
A stirring rendition of You'll Never Walk Alone - a song both teams have adopted as their football anthem - before kick-off set the tone for the game, with a special atmosphere playing its part in an encounter that was tense and, at times, thrilling.
A sold-out Westfalenstadion united to sing You'll Never Walk Alone before the match
Klopp gets his tactics right
Some eyebrows were raised before kick-off when Klopp decided to start Origi instead of England international Daniel Sturridge.
Origi had not scored since 14 February, while Sturridge appeared to be finding his form after returning from injury with three goals in nine games.
The 20-year-old was nothing more than a spectator in the opening stages as Liverpool soaked up some early pressure, but Klopp said before the game that he had given him the nod because of his pace and power - and both were on show when the visitors took the lead.
He controlled captain Jordan Henderson's slightly awkward ball superbly before turning towards goal and steering a low strike into the far corner.
From then on, Origi held up the ball excellently to relieve some pressure on a defence that unsurprisingly came under some heavy pressure at times.
In Pierre-Emerick Aubameyang, Dortmund possess one of the most lethal finishers in Europe, but Dejan Lovren and Mamadou Sakho stood up to the task of nullifying the 33-goal Gabon striker.
He was limited to one real chance, an effort from close range that was blocked by Lovren, while Sakho threw his body in the way of Henrikh Mkhitaryan to deny what was an almost certain goal.
Eyes on Klopp for 'El Kloppico'
The 48-year-old German is fondly remembered for his achievements at the club he managed for seven years from 2008, having led them to two Bundesliga titles as well the final of the Champions League in 2013.
Around 20 television crews and some 100 journalists were in attendance for his pre-match news conference, while one German TV had been planning a 'Klopp Cam', which would have focused on the Liverpool boss for the full 90 minutes - although that idea was ultimately shelved.
When Klopp emerged before the match, his name was chanted by the home fans but the Liverpool boss was evidently keen to make sure he was not the centre of attention.
After acknowledging the home fans with a clap, he switched focus firmly to his own players warming up and there was no doubt about where his allegiances lie when he thumped his chest and roared in delight after Origi's goal.
Despite the away goal, the tie is far from over. Dortmund have scored in each of their previous six away games, including two goals at Tottenham in the previous round of the competition.
Improving their defending at set-pieces will also be key for Liverpool. Hummels was left unmarked to head in the simplest of goals from a corner, meaning the past three goals Liverpool have conceded in the Europa League have been from set-pieces.
History is on Liverpool's side, though. They have progressed from 11 of their past 12 European ties in which they have drawn the first leg away from home.