Selma Panengstuen, Caitlin Dijkstra, Victoria Pelova and Shekeira Martinez all started against the Liga F side with Kirsty Hanson entering the action off the bench in the second half.
Martin Ho's team dominated in the Spanish heat - temperatures soaring into the mid-30s in the second half - but had to wait until four minutes from time to get the winner the performance deserved. Eveliina Summanen picked out Gaupset's run in the left channel and the Norway international did the rest, cutting onto her right foot, nutmegging Jujuba Cardoza and curling into the far corner.
It was one-way traffic in the first half as Grenada's goalkeeper Laura Sanchez kept us at bay.
The early threat came from our left with Julie Blakstad and Matilda Vinberg prominent. Vinberg's overlap led to the first effort sliced wide by Pelova before Blakstad drove into the box and rolled into Vinberg, Sanchez blocked her effort, the ball spun towards goal only for Teresa Merida to get back in time.
Our press was also prominent and led to the next two chances. Olga Ahtinen's pressure saw Olivia Holdt's shot deflected wide - Vinberg's inswinging corner cleared off the line - before Matilda Nilden robbed Daniela Minambres and drilled low at goal, her shot deflected and kept out by Sanchez, who got up to block Martinez's follow up.
Grenada's only real chance came on the counter as Miku Kojima's early pass opened us up with Amanda Nilden facing a two on one, however, Barbara Lopez went for goal early and fired straight at Panengstuen from 30 yards despite having Molina in support.
The pattern continued in what turned out to be a stop-start second half with a number of substitutions and series of player treatments required for the home side.
Chances continued to come our way - Blakstad latched onto Gaupset's pass, clipped past Sanchez but inches wide. Martinez then lobbed Sanchez, only to see the goalkeeper get the slightest touch and the ball float just wide. Gaupset was off target from 20 yards and then Hanson might have scored with her first touch in a Spurs shirt but dragged her effort wide from the edge of the box.
With time ticking away and no little hint of frustration creeping in, Gaupset provided the decisive action - Summanen picked out the pass, Gaupset cut into the box, beat Jujuba Cardoza and planted her shot beyond substitute goalkeeper Alba Ibanez.
There was still time for two more chances - Drew Spence picked out Araya Dennis, her shot tipped over by Ibanez, and in the last action, Ibanez was off her line quickly to save at the feet of Hanson.
Line-ups
Grenada CF Femenino: Sanchez, Merida, Minambres, B Lopez, Molina, Sanmartin, Evans, Kojima, Gil, Jujuba Cardozo, Perea. Substitutes: Ibanez, J Lopez, Martos, Fon, Carlota, Vero, Elena.
Spurs: Panengstuen (Kop 46), Morris (Dennis 70), Dijkstra (Koga 46), A. Nilden, Holdt (Gaupset 46), Vinberg (Pierre 70), M. Nilden (Hanson 70), Martinez (Summanen 70), Ahtinen (Spence 70), Pelova (Tandberg 70), Blakstad (Hills 75). Substitutes (not used): Jackson, Sombath.
Match stats
Goals: Spurs - Gaupset 86.
Yellow cards: Spurs - A Nilden,
Referee: Mireya Lopez Reche.
Venue: Pinatar Arena, Murcia.
Weather: Hot, 32 degrees at kick-off.