The Belgians had dominated the first 90 minutes but a barrage of efforts were thwarted by an inspired Howard, whose resistance was finally broken in the 93rd minute when midfielder Kevin De Bruyne buried an angled shot from seven metres.
In a pulsating extra time, substitute striker Romelu Lukaku powered home a second in the 105th minute after a clever De Bruyne pass, before Julian Green's neat volley ensured a tense finish in which Clint Dempsey almost sneaked an equaliser.
"We had about 15 chances, we were controlling the game I think it is largely deserved even though at the end we conceded one," Belgium coach Marc Wilmots told reporters.
"My players reached their limits to qualify. I said fresh legs would be important today and that is how it was. Now the whole of Belgium can celebrate."
Wilmots's side had arrived at the Fonte Nova arena with three successive wins in Brazil, although the team's lacklustre performances had raised doubts about their ability to go deep into the tournament.
Much sharper attacking intent against the U.S. would have pleased their fans, although congratulations would have turned to recriminations had U.S. substitute Chris Wondolowski not horribly skewed wide from five metres in the last seconds of normal time.
"It was real drama, a thriller, we had enough possibilities to equalise the game or even put it away earlier," said U.S. coach Juergen Klinsmann, whose side had seen off Ghana and Portugal to qualify for the last 16.
"It was a game that just went to the extreme."
In a sign of things to come, striker Divock Origi was in on goal inside 40 seconds, only to see his low drive parried away by Howard and more chances stacked up as the U.S. were forced to set up camp outside their own box and look for quick breaks.
The lively De Bruyne should have done better after 23 minutes having found space in the box, while Clint Dempsey's tame volley was the Americans' only chance of the first half.
The Belgian onslaught cranked up a gear after the break with the scrambling U.S. defence grateful to see Howard save efforts by Dries Mertens and Jan Vertonghen before Origi's header hit the crossbar.
A Kevin Mirallas run and shot, two more stinging Origi drives and a Vincent Kompany effort were all dealt with by Howard, who faced 38 attempts on his goal during the game.
With extra time moments away, Wondolowski missed a glorious chance that the Americans would come to rue.
All Belgium's goals in Brazil had been scored after the 70th-minute mark and their persistence finally paid off in extra time when De Bruyne was given time and space to turn and fire home after Lukaku raced down the right wing to cross.
The midfielder soon returned the favour, teeing up the frontman to power home a second and spark a thrilling finale.
Roared on by deafening chants of 'USA', Klinsmann's side summoned the energy to raise American hopes.
Midfielder Green met a delicately chipped Michael Bradley pass to volley home two minutes later and Dempsey almost forced penalties, only for his clever set-piece effort to be foiled by Thibaut Courtois.
"It was heart-breaking. We left it all out there but we lost to a really good team. It hurts but hats off to Belgium they were fantastic," Howard said.
"Sometimes when you give your best it doesn't come off."
The win sealed Belgium's first World Cup quarter-final appearance since 1986 and they will next play Argentina on Saturday.