The Gunners were trailing 2-1 from the first leg and had to do without injured captain Cesc Fabregas at the Emirates Stadium.
However, Arsene Wenger's young team rose to the challenge as a first-half brace from Bendtner - who missed a string of chances against Burnley here three days ago - was followed by fine goals from Samir Nasri and Emmanuel Eboue before the young Dane slotted home a penalty in the final minute to secure a place among the last eight of Europe's elite once again.
The result also broke Arsenal's unwelcome near 32-year run of failing to overturn a first-leg deficit in a European tie since beating Hajduk Split in 1978.
Arsenal had started brightly, with Nasri's shot blocked in the six-yard box before the lively Andrey Arshavin saw his far-post header saved following a deep cross by Bacary Sagna.
The Gunners went ahead after 10 minutes.
Goalkeeper Manuel Almunia, standing in as captain, booted a long clearance up field, which Arshavin nodded back for Nasri.
The Frenchman knocked a quick pass between the centre-backs for the Russian to run onto.
In tracking back, Porto defender Fucile clattered into his keeper, and Bendtner was first to react, stabbing the loose ball into the empty net.
Helton needed treatment for a head injury, caught accidentally by Arshavin, but after a short delay, the Brazilian was able to continue.
With the tie now level, Arsenal looked to go ahead, as Abou Diaby hooked a 12-yard effort over.
Porto continued to pass the ball crisply, but were undone again on 25 minutes.
Fucile's poor clearance was latched onto by Arshavin.
The diminutive Russian weaved into the left side of the penalty box, before pulling the ball back along the six-yard box to Bendtner, who, despite what he showed here on Saturday, just could not miss.
Arsenal were somewhat fortunate when Brazil forward Hulk - who is serving a four-month domestic ban for a fracas with a steward after a match against Benfica - capitalised on a mistake by right-back Gael Clichy and cut in towards goal, but stabbed his shot into the arms of Almunia.
Helton touched Bendtner's curling effort around the post.
From the corner, Diaby's downward header was clawed away by the Porto keeper, who also denied Nasri in stoppage time.
There was a scare at the start of the second half when Tomas Rosicky gave the ball away to Hulk, but fortunately for the Czech winger, the Porto striker was again off target.
Almunia then had to be on his toes as Falcao drove in a low snapshot, which the Spaniard gathered at the second attempt.
Rosicky, who appeared to be carrying a knee problem, was replaced by Emmanuel Eboue just before the hour as Wenger looked to breathe new life into his side.
Arsenal, though, were anything but convincing at the back - with Nasri clearing a header from substitute Cristian Rodriguez off the line.
The Frenchman, however, then settled any nerves with a quite brilliant individual goal after 64 minutes.
After collecting the ball just outside the area on the right, Nasri twisted past three Porto defenders before drilling a low shot in off the far post from the tightest of angles.
Safe passage to the quarter-finals was assured three minutes later when, following a Porto corner, Arshavin broke quickly down the left, holding the ball up before releasing the on-rushing Eboue, who rounded the keeper and stroked in a fourth.
Porto had lost here 4-0 in the group stages last season, which was their worst away reverse in the competition.
And over the final 20 minutes, it was a case of damage limitation for Jesualdo Ferreira's men.
Nasri was given a well-earned rest as he left the field to a standing ovation, replaced by defensive midfielder Denilson.
Fellow substitute Theo Walcott looked lively after he came on for Arshavin, and Bendtner headed a good hat-trick chance wide.
Eboue was brought down for by Fucile to concede a penalty in the final minute - which the young Dane stroked into the bottom corner to collect the match ball.