Summer signing Nmecha, a free transfer from Wolfsburg, held his nerve to bury his 84th-minute spot kick - just six minutes after stepping off the bench to replace Joel Piroe - to clinch the hosts a deserved 1-0 victory at Elland Road on Monday night.
"I was a bit worried. He's on the sheet for taking penalties, but as a substitute, he needed more touches and I was thinking of stepping in there and taking the ball off him," Leeds manager Farke said.
"Thank God I calmed myself down and he prove me right with an ice-cold finish. He deserves today to be in the spotlight after not the easiest of periods in these last few years."
Germany international Nmecha struggled with injuries in his last two seasons in the Bundesliga and has gone under the radar since becoming the first of Leeds' eight summer signings so far.
Leeds dominated for long periods against Everton, who handed a debut to on-loan England international forward Jack Grealish, and Farke added: "Overall, in a tight game, a hard-fought win, but also a well-deserved win.
"I think for a newly promoted side, it was an outstanding performance.
"We were very dominant in the first half. Nearly 70 per cent possession, 12 shots, I think 21 in total."
Everton boss David Moyes admitted his side's display did not warrant a point, but did not believe the home side should have been awarded a penalty.
Toffees skipper James Trakowski appeared to lean into Leeds debutant Anton Stach's deflected shot and although his arms were by his side, the ball struck his upper arm and referee Chris Kavanagh's decision was upheld by VAR.
"I'm obviously going to say it's not a penalty and I believe it's not a penalty," Moyes said.
"I actually think the referees have had a poor weekend in the opening weekend.
"I think there have been a lot of (poor) decisions and tonight's is another one."
Everton, who lost only three of their last 18 league games last season following the return of Moyes for his second spell in charge in January, rarely threatened and slipped to their first defeat at Elland Road in six league matches, stretching back to 2001.
When asked about his side's display, Moyes added: "We improved a bit in the second half, but I wasn't impressed by how we played at all."
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