A much-changed Forest side were under the pump against the Primeira Liga leaders and fell behind to William Gomes' early strike.
But they were gifted a leveller by hapless Porto defender Martim Fernandes, who sent a 25-yard back-pass into his own net.
Porto were in controlled possession and right-back Fernandes passed back to his goalkeeper Diogo Costa, but did so without looking and sent the ball wide of the No.1 and into an empty net for a comical own goal.
Forest had a goal disallowed in the second half, but were reliant on goalkeeper Stefan Ortega who made a string of important saves to ensure they head back to the City Ground for next week's second leg on level terms.
Importantly, they were able to get the result while still being able to keep a number of their important Premier League players fresh ahead of Sunday's vital clash with Aston Villa.
If Forest can get the job done on home soil next week, it looks highly likely as if Villa will be waiting in the semi-final after they won their first leg 3-1 at Bologna.
Indeed, Villa were clearly on boss Vitor Pereira's mind.
Forest were playing their first European quarter-final since losing to Bayern Munich in the UEFA Cup 30 years ago, but the Portuguese made it clear what his priority is, making nine changes to the side that beat Tottenham in the Premier League before the international break.
One of them was the returning Chris Wood, who was playing his first match since mid-October following a knee injury.
In Bologna, Ollie Watkins' second-half brace helped give the Villans a comfortable lead heading into next week's second leg at Villa Park.
Bologna had not tasted defeat in the competition since the opening game of the campaign where Villa beat them 1-0, and Unai Emery's side proved to be their nemesis once again to end their 11-game European unbeaten run at the Renato Dall'Ara Stadium.
Santiago Castro's strike was ruled out by VAR for offside before Lewis Ferguson hit the crossbar moments later for Bologna.
The Premier League side managed to hold on and punished the Italian side with goals either side of half time from Ezri Konsa and Watkins.
Bologna breathed life into the contest when Jonathan Rowe scored late until Watkins' stoppage time effort restored their two-goal advantage.
In other results, Freiburg beat Celta Vigo 3-0 while hosts Braga and Betis drew 1-1.Â
In the Conference League, Jean-Philippe Mateta's first goal since New Year's Day inspired Crystal Palace to a 3-0 victory over Fiorentina in their first leg of their quarter-final.
The France striker, who started for the first time since January, opened the scoring with a 24th-minute penalty, and Palace took a two-goal lead when Tyrick Mitchell recycled the rebound from a Mateta effort seven minutes later.
Ismaila Sarr ensured Palace will travel to Tuscany with a comfortable advantage when he nodded home their third on the stroke of time.
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