Filip Kostic's goal from an offside position was allowed to stand in the 36th minute and Sebastian Rode levelled the tie at 4-4 just over 20 minutes from the end to send the Bundesliga side through to the semi-finals.
Benfica had established a strong position with a 4-2 victory a week ago but were unable to muster a response after the linesman's mistake in the first half left their lead looking tenuous.
Substitute Eduardo Salvio hit the upright with a late chance to put the Portuguese outfit back into the box seat, but Frankfurt held firm to book a final-four meeting with Chelsea.
Benfica defended stoutly for much of the first half and only an officiating error eroded their two-goal advantage.
Replays showed wing-back Kostic to have been comfortably offside when he turned in the rebound from a Mijat Gacinovic shot that struck the base of the left post.
Bruno Lage's protestations resulted only in the Benfica head coach being sent to the stands before the break.
His team briefly displayed a greater sense of urgency after the interval as Haris Seferovic worked Kevin Trapp with a close-range header.
The hosts, though, always appeared a threat going forward and the crucial strike eventually came from an unlikely source, midfielder Rode side-footing into the bottom-right corner from the edge of the area.
The entire complexion of the contest almost swung towards the death, but winger Salvio could only find the outside of the post from Alejandro Grimaldo's cross as Benfica fell narrowly short of advancing.
What does it mean? Benfica rue avoidable mistake
The implementation of technology emerged as a major talking point from a dramatic Champions League encounter at Etihad Stadium on Wednesday and Benfica will rightly wonder why they did not receive the same level of officiating support as Tottenham.
Rebic makes key contribution
Benfica generally did a commendable job of restraining Eintracht's talented attackers, but the tireless Ante Rebic managed to make a major impression on the match when it mattered most. It was his deft touch that created the space for team-mate Rode to land the knockout blow.
Seferovic flounders on Frankfurt return
Seferovic endured three fairly lean seasons with Eintracht before moving to Benfica in 2017. A largely anonymous showing that included one squandered opportunity and little else will have evoked unhappy memories of that period.
Frankfurt now face Chelsea in the semi-finals, with the two legs taking place on May 2 and May 9. They must first visit Wolfsburg in the Bundesliga on Monday, the same day as Benfica's home game against Maritimo in the Primeira Liga.