It was Everton's only second league win on the road this season, having last won an away league game in August against Brighton.
"It's about the team performance. To win 2-1 away from home -- it feels like forever since we won away from home," said goalkeeper Jordan Pickford, who made stunning saves in the second half.
"It's about commitment, attitude, team spirit. That's what gets you over the line."
Holgate nodded in a rebound to put Everton back in front at the half-hour mark after Leicester striker Patson Daka cancelled out Mykolenko's early opener.
Earlier, Mykolenko gave Everton the perfect start, firing a stunning left-foot volley from the edge of the box in the sixth minute after Alex Iwobi clipped a ball in from the right.
But Leicester, looking to bounce back after their Europa Conference League exit, recovered quickly with Daka levelling in the 11th minute following a defensive mix-up.
Leicester created many chances in the second half, especially through substitute Harvey Barnes, but Pickford produced another inspired performance.
Everton have four matches left this season, one more than all the teams below them. They travel to Watford on Wednesday.