Frank Lampard's side moved up two spots to 16th on 35 points, one point above the drop zone, as Leeds United slipped to 18th after a 2-1 defeat at Arsenal. Watford and Norwich City have already been relegated to the second-tier Championship.
It was Everton's only second league win on the road this season, having last won an away league game in August against Brighton & Hove Albion.
"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.
The unmarked Holgate scored from close range after Richarlison's header was beaten away by Leicester goalkeeper Kasper Schmeichel following a corner.
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.
England international Pickford tipped away Nampalys Mendy's curling effort and denied Barnes twice as Everton held on to their lead.
Everton have four matches left this season, one more than all the teams below them. They travel to Watford on Wednesday.