Gill ended Friday's first day on 127 not out in his maiden Test as skipper and Jaiswal made 101, becoming the first opening batter from Asia to hit a hundred at Headingley.
Gill was key in the two big partnerships, putting on 129 with Jaiswal for the third wicket that tore hope from England, and then controlling an unbeaten 138-run stand with the unpredictable Rishabh Pant, who ended 65 not out at stumps.
It all followed India being asked to bat first by England. India would have bowled first, too, Gill admitted, but it turned out to be a good toss to lose.
England, despite a weakened pace attack, hoped for some early movement off a green-tinged pitch, backed up by recent history, with the last six Test winners at Headingley all bowling first.
But movement was limited and England got their lines and lengths wrong, toiling all day as the three pacers went for 4.3 an over. Captain Ben Stokes, with only one competitive game this year, was the pick with his 2-43.
India would have won the morning easily but Lokesh Rahul and debutant Sai Sudharsan were both out to edges on the stroke of lunch.
Joe Root at first slip took his 209th catch in Tests, one shy of Rahul Dravid's record for a non-wicketkeeper.
Jaiswal, on 42 at lunch, and Gill then rebuilt the innings in style.
Jaiswal, who didn't offer a chance, reached his fifth hundred, and third against England, just before tea.
But in the 80s he began to show apparent painful cramp in his right forearm. After on-field treatment he carried on, and stayed in the 90s for only six balls.
He celebrated his hundred by running around the pitch swinging his helmet and punching the air. He has 813 runs against England in 10 innings.
Jaiswal was out straight after tea for 101 off 159 balls, bowled by a beauty from Stokes.
Pant came in and slapped the second ball he faced from Stokes back over the captain's head, drawing a big smile from the England skipper.
But Pant and Gill restrained themselves and India's scoring dropped below four an over as spinner Shoaib Bashir was allowed to hold down one end.
As the bowlers tired, they lifted the run rate. Gill's 14th boundary earned his hundred off 140 balls, and he screamed in delight, bowing to the dressing room.
A six over fine leg sent Gill past 2000 test runs, and Pant followed him, passing 3000 Test runs with his 22nd half-century off 91 balls. He then hit two late sixes to rouse India's happy fans.
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