Mauricio Pochettino's men held a commanding lead ahead of their trip to Signal Iduna Park and, although the hosts piled on the pressure in the first half, Hugo Lloris proved a reliable last line of defence before Harry Kane ultimately put the tie to bed.
Dortmund did everything they could to breathe life into the second leg in the opening period, but they came up against an inspired Lloris, who made a string of vital saves, most impressively to thwart Mario Gotze and Julian Weigl.
Spurs then took full advantage just after the interval, Kane confidently dispatching a one-on-one to become the club's all-time leading scorer in European competition with 24 goals.
Dortmund spent much of the early stages in the ascendancy and looked destined to take an 11th-minute lead when Marco Reus was released into the area, but Jan Vertonghen produced a perfectly timed last-ditch tackle.
A rare chance fell to Spurs on the break 20 minutes later, but Son Heung-min missed the target under pressure from Marius Wolf with just Roman Burki to beat.
Spurs came under siege soon after, but they remained unscathed as Lloris produced fine saves to deny Weigl, Gotze and Jadon Sancho all within a few minutes.
Any hope built up by Dortmund's first-half performance swiftly vanished after half-time, however, as Moussa Sissoko fed Kane with a first-time pass and the England striker confidently fired past Burki.
Dortmund crafted one final chance a minute from time, but the outcome summed up the match, as Lloris blocked Paco Alcacer's effort, enabling Spurs to cruise into the last eight.
Pochettino will surely focus his attentions on their European campaign now that their Premier League title tilt is all-but over.
What does it mean? Dortmund's revolution has a way to go yet
Dortmund have been widely lauded for their youth policy and vibrant football this season, but against Spurs their inexperience showed in both legs, even if their style caught the eye on occasion.
Lloris shows his class
It is fair to say Lloris has attracted criticism at times this season for costly errors, but he was at the top of his game on Tuesday, particularly in the first half when he thwarted Gotze, Sancho, Weigl and Alcacer.
Reus unable to spark Dortmund comeback
As one of few genuinely experienced campaigners in this Dortmund team, Reus was expected to lead the way. While he was by no means terrible, he failed to provide the creative spark they needed and was withdrawn towards the end.
Key Opta facts
- Tottenham have qualified for the quarter-finals of the Champions League for just the second time in the club's history.
- Spurs have beaten Borussia Dortmund in each of their four Champions League meetings.
- Having lost just two of their first 12 home European contests against English sides (W6 D4 L2), Dortmund have now lost back-to-back such matches (both against Spurs).
- Twelve of Tottenham's 13 goals in the Champions League this season have come in the second-half (92 per cent).
- Harry Kane is just the second Englishman to score at least five goals in multiple Champions League campaigns after Steven Gerrard, who did so in 2007-08 and 2008-09.
It is back to league action for both teams on Saturday as Dortmund host Stuttgart, hoping to bounce back from defeat to Augsburg, while Spurs travel to Southampton having not won any of their last three in England's top flight.