The Argentina international was first brought to the ground by Toffees defender James Tarkowski in the early stages of the match, with Bruno Fernandes stepping up to hand the home side a 12th-minute lead.
Garnacho was then fouled inside the box by Ben Godfrey in the latter stages of the first half, and Marcus Rashford netted on the second occasion to double Man United's lead on the afternoon.
The forward is the first Man United player to win two penalties in the same Premier League match, while it is also the first time that the 20-time English champions have had two different penalty scorers in the first half of a Premier League fixture, with Fernandes allowing Rashford to take the second spot kick.
Garnacho has been a vital player for the Red Devils this season, with the 19-year-old making 37 appearances in all competitions, scoring seven goals and registering five assists in the process.
"Garna and Rashy, they are almost unstoppable, when they have the moments in behind. Garnacho is progressing very well and we are very pleased with this," Man United head coach Erik ten Hag told reporters after the two-goal success over Sean Dyche's side.
"His contribution today was huge, with his runs, his dribbles. It's attractive as well for the audience, obviously we want to show something.
"I think [they were] well-deserved penalties. He gave us space to get in behind and Rashy also, when he gets in those positions, is difficult to stop."
Ten Hag also opened up on the importance of the win, which moved the Red Devils to within three points of fifth-placed Tottenham Hotspur in the Premier League table, albeit having now played two games more than the capital outfit.
"It was a win, it was important. The team fought, with team spirit, and created a lot of chances because Everton were very brave in the high press," said the Dutchman.
"We could have done with a bit more calmness and composure maybe. We needed this win. We wanted to keep the pressure on the teams above us.
"We had, this year, one poor performance, probably against Fulham. Most of the other games, we won. We need wins. We know we need to catch up after a difficult autumn.
"We're still in the hunt but it depends on them. We have to win our games but we know, in the Premier League, everyone is killing everyone. They will also drop points so, if we do our job, we can close [the gap]."
Man United will now switch their attention to the FA Cup, with Liverpool heading to Old Trafford in the quarter-finals of the competition next Sunday.