World number 68 Giraldo beat Albert Montanes - a man positioned nine places above him in the rankings - 6-1 7-6 (7-3) at the hard-court ATP 250 event in Auckland.
The Colombian, who reached the semi-final of this tournament in 2011, got off to a flying start on Monday by wrapping up the first set with the minimum of fuss, before Spaniard Montanes fought back in the second only to be beaten in the tie-break.
Giraldo's reward for his victory is a second-round tie against either eighth seed Robin Haase or Guillermo Garcia-Lopez.
Zeballos' match followed a similar pattern to Giraldo's, as the Argentinian overcame Filippo Volandri 6-0 7-6 (7-1).
The 28-year-old cruised through the opening set - winning 83 per cent of first-serve points in the process - but was then forced to battle his way to the last 16 as Volandri responded well in the second.
Zeballos is now likely to face fifth seed Philipp Kohlschreiber, with the German - who was beaten by David Ferrer in last year's final - playing the unseeded world number 61 Pablo Carreno Busta in the first round.
American wildcard Sock, meanwhile, will meet veteran second seed Tommy Haas in round two after he wrapped up a 6-3 6-4 win over Adrian Mannarino in a little over an hour.
The 21-year-old fired down 11 aces during the match to beat the Frenchman, who is ranked 71 in the world - 29 places higher than Sock.