The Australian served for the match at 5-4 in the second set, having yet to face a break point for the encounter in Bogota, only for Cabal to manufacture one chance and take it.
Groth then wasted three match points in the second-set tie-break, and blew a break lead in the decider before prevailing 6-3 6-7 (10-12) 7-6 (7-4) over the Colombian wildcard.
The match lasted two hours 21 minutes, and Groth will have to back up from the lengthy contest against top seed Richard Gasquet, who enjoyed a bye past the first round.
Earlier, there was better news for the local contingent at the hard-court tournament, as fifth seed Alejandro Falla and sixth seed Alejandro Gonzalez progressed.
Falla, runner-up at the ATP 250 event in 2013, was too good for Japan's Tatsuma Ito 6-3 6-2, and the left-hander set up a second-round date with Australian wildcard Bernard Tomic.
Gonzalez won his first match at his home event at the second time of asking, progressing past Russia's Evgeny Donskoy 7-5 3-6 6-4 in two hours 20 minutes.
Next up for Medellin native Gonzalez is Dutchman Thiemo de Bakker, who saved a match point before seeing off American qualifier Kevin King 2-6 7-6 (7-4) 6-3.
Serving at 5-6 in the second set, De Bakker was one point from defeat only for him to deny King a maiden tour-level victory as he won through in just over two hours.
Israel's Dudi Sela was a 6-4 6-2 victor over Argentina's Facundo Arguello in under an hour, to set up a second-round date with defending champion and second seed Ivo Karlovic.