Kevin De Bruyne and substitute Romelu Lukaku's scored two goals in the first half of extra-time before teenager Julian Green, who came in the second period of extra-time, reduced the deficit for USA with his first touch.
Belgium will now face two-time champions Argentina in the quarterfinals in Brasilia on Saturday after the La Albiceleste's rode on a goal from Di Maria deep into additional time to beat Switzerland 1-0 in Sao Paulo in another last-16 match.
Lukaku then doubled the lead in the 106th minute, this time with an assist from De Bruyne, to put the game beyound USA's reach.
But USA did not give up without a fight and fired back when Bayern Munich teenager Green pulled one back with a first-time volley from a Michael Bradley pass.
Belgium had to go through some nervous moments in the final 10 minutes of the match but did enough to hold on to their slender lead.