The USA booked its spot in the last eight after beating Paraguay 1-0 in the last match of the group, with Dempsey scoring the lone goal.
Gyasi Yardes setup his teammate from a well-taken cross, as the Seattle Sounders man made no mistake in converting in the 27th minute.
DeAndre Yedlin was sent off early after the break, but the USA held on for the remainder of the match, securing all three points in a crucial encounter.
A pair of bookings in a matter of minutes saw Yedlin sent off, but Paraguay was unable to make the most of its numerical advantage.
Paraguay struggled to break down the Copa hosts in the second half and bowed out in the group stage for the first time since 2001.