The Magpies went into the fixture with a healthy 6-1 lead from the first leg clash in Azerbaijan last week and went further ahead only four minutes into the second leg with Sandro Tonali tapping in a rebound.
Joelinton doubled the Toons lead just two minutes later with a neat volleyed finished from a lovely Harvey Barnes cross.
The visitors attempted to fight back, with Elvin Cafarguliyev forcing Aaron Ramsdale into a save with his foot from a tight angle while Camilo Duran had his attempt tipped over the bar.
Duran eventually pulled one back for Qarabag after he outran Burn to slot a low neat finish past Ramsdale.
However, Newcastle responded almost immediately with Sven Botman powering a header in from a Kieran Trippier corner-kick to restore the two-goal lead.
Qarabag were awarded a penalty in the 57th minute after a handball from Burn in the box, but Marko Jankovic's spot kick was initially saved by Ramsdale with Cafarguliyev Squeezing in the rebound.
Both teams had chances to score more goals but could not convert, with the goalkeeper's making some good saves as the match ended 3-2.
"It's a massive achievement from us to get to the last 16," said Eddie Howe after the 9-3 aggregate win.
"Nothing was given to us at the start of this Champions League run. We've really enjoyed it, and then coming to the pressure game, the double-header, I think we've navigated that really well.
"I don't care who you play at this stage, it's a tough draw, and we did the majority of the work in the away game. We've come through really well."
The Tyneside club will either face Chelsea or Barcelona in the Round of 16, with the draw set for Friday, February 27.
Liverpool FC will discover their route to the Champions League final on Friday, when the draw for the last 16 maps out not only their next opponents but also their potential path all the way to May's showpiece final.
Aurelien Tchouameni has described Real Madrid's progression to the UEFA Champions League last 16 as a victory for "everyone who is against racism" after a tense 2-1 win over S.L. Benfica at the Santiago Bernabeu on Wednesday night.
A frank dressing-room meeting proved the catalyst for Arsenal's emphatic north London derby victory, with Viktor Gyokeres revealing the squad "let it all out" before dismantling Tottenham 4-1.
Cristian Chivu has declared that belief and courage would be key as Inter Milan prepared to face Bodo-Glimt tonight at San Siro, needing to overturn a 3-1 first-leg deficit in the Champions League play-off.
