The final was part of a quick-fire double-header between the clubs – they played a cup semi-final earlier in the week ending in a 2-0 victory for Whites – and Easthouses handled the big game pressure better, notching the only goal of the game four minutes into extra-time.
The solitary goal arrived in somewhat fortuitous circumstances for the Midlothian side, as Whites' otherwise flawless goalkeeper, Hamish Innes, appeared to lose sight of an airborne ball amid the glare of the floodlights and allowed Liam Dunbar to seize the loose ball and capitalise fully.
Easthouses coach Kevin McCallion felt sympathy for his opponents' goalkeeper, but felt his own side were worthy winners.
"Their goalkeeper is probably the best I've seen all season at all levels," he said. "He was confident, but he made one mistake and we scored – it's always going to be difficult under the floodlights. The ball came through the lights and, nine times out of ten, the boy would have saved it, but he spilled it to the left and Liam Dunbar tucked it in.
"Liam's a boy who doesn't get as much time on the pitch as some others, so I'm delighted for him. He was the supersub for us!"
Only moments had lapsed after the match had entered added time when Easthouses scored, but in regulation time the tie was as neck-and-neck as to be expected.
"It was a tight game," said McCallion. "We were pressing in the first half but, come the second half, it became tighter. After we scored, City had a couple of chances to equalise, but other than that neither side gave much away.
"Liam Thomson, Michael MacFarlane and Dayle Orr were outstanding and we got a good run from Thomas Prior before he got an injury."
Easthouses have made healthy progress in cup competitions this year, including three wins in the Scottish Cup before a fourth-round exit. After difficulty with fixture scheduling recently and having secured their first piece of silverware, the club will switch more of its attention to climbing the Third Division league table.
"We were a bit below par against City, to be honest," said McCallion. "We did come up to speed eventually, but it's been quite a hard winter for the boys.
"We're now catching up in the league and, whereas at the start of the season all the teams were finding their feet, every game is a hard one now."
Meanwhile, Whites secretary Tom Dumayne was disappointed that his side could not replicate the result of the clubs' previous clash, but paid tribute to the efforts of his players. He said: "On the previous Sunday, we played Easthouses in another cup and won that game 2-0.
"I wouldn't say we were confident going into the cup final, though. We just knew it would be a very close game.
"The pitch didn't do us any favours – it wasn't a good pitch for a final. It was bumpy and has been played on quite a lot and, with the rain, it was very heavy. Obviously, it's the same for both sides, but I think it suited them more than us, because we are quite a lightweight team.
"Our defence was very strong – it's a good part of the team. We are an attacking force as well, but we were pinned back in large parts of the game. Some chances fell to us, but again on the pitch the ball didn't bounce as you'd expect and ran away from us on a few occasions.
"Because of the surface, we knew there was going to be a mistake from either our side or theirs. Easthouses played the game in a good spirit, though, as we did, so there's no bad feelings."
Easthouses: Chris Ramage, Alex Caterer, Sean Knight, Kristopher Snedden, Dayle Orr, Kieran Hosie, Michael MacFarlane, Thomas Prior, Ross Moir, Connor McCallion, Euan Russell, Lindsay Birrell, Liam Dunbar, Liam Thomson, William Hutchison, Lewis Hogarth.
Edinburgh City Whites: Hamish Innes, Bobby Findlay, Craig Paterson, Connor Brownlie, Alex Falconer, Ross Edmonds, Callum Miller, Cameron Whyte, Danny Caira, Charlie Donnelly, Dale Maitland, Cameron Stark, Fraser Dumayne, Jamie Appleton, Alex Murphy, Robert McGrath.