Adam Lallana scored his first goal for Brighton in the 10th minute and was close to adding a second when he hit the post with a header before half-time.
An impressive Brighton also had a goal disallowed in the first half but Leicester improved after the break and Kelechi Iheanacho equalised in the 62nd minute.
The hosts looked nervy late on and Amartey stooped to head in from close range in the 88th minute.
It was tough on Brighton, who remained down in 16th position, three points above the relegation zone after three narrow defeats in a row.
Injury-hit Leicester, now unbeaten in 10 league away games, moved two points ahead of Manchester United into second place with 53 points from 28 games. United face runaway leaders Manchester City on Sunday.
"The first half was not good enough at all. It was not difficult to do better in the second half," Leicester midfielder Youri Tielemans, who set up the equaliser, said.
"It feels like a very big result."
With injuries to key players such as James Maddison taking their toll, there have been signs that Leicester were beginning to falter having looked on course for a top-four finish.
Leicester were denied an equaliser by a superb save by home keeper Robert Sanchez from a thunderous strike by Sidnei Tavares but Brighton were agonisingly close to making it 2-0 when Pascal Gross whipped in a free kick and Lallana's glancing header hit the far post.
There was still time before the break for Maupay to produce an exquisite finish from Lallana's pass but Leicester were saved by an offside decision.
Brighton were to rue those opportunities just past the hour mark when Tielemans's sliderule pass took out the Brighton defence and Iheanacho dinked a finish past Sanchez.
Sanchez then flapped at Marc Albrighton's corner and Amartey was in the right place to head home.
Leicester manager Brendan Rodgers admitted his side had started slowly.
"We needed to be more aggressive. Once we did that we had more energy on the ball," he said. "The mentality has been incredible -- to be up there challenging with 10 games to go."