Knott and Maia Bouchier struggled to get the ball away amid some smothering Surrey pressure, scoring just 32 in the six-over powerplay.
They were 57-0 at the start of the 10th over when Bouchier sliced to Bryony Smith off Alice Capsey to depart for 33 from 25 balls.
Phoebe Franklin ran round from long-on to oust Freya Kemp with a fine low catch inside the rope, and took another off Dani Gregory (2-22 off four overs) falling to her left at mid-off to account for skipper Georgia Adams and leave her side 95-3 in the 15th over.
Knott brought up the third slowest half-century of the 31 made in the competition this season from 44 balls, including five boundaries, but picked up the pace to move into the 60s before Davidson-Richards took a fine diving catch just above the turf to dismiss Rhianna Southby for 10 and leave the Hawks 122-4 with 18 balls remaining.
Abi Norgrove played positively when she joined Knott, striking four boundaries in a 10-ball 24 not out, while Knott finished unbeaten on 74 from 61 balls with eight fours.
England’s Linsey Smith picked up a wicket with her first delivery as Surrey skipper Bryony Smith chipped tamely to opposite number Adams to leave Surrey 3-1 from seven deliveries.
Wyatt-Hodge clubbed Linsey Smith over long-off for the only six of the match, but Alice Capsey chipped Lauren Bell straight to Knott at mid-wicket, and Norgrove took a stunning catch diving high to her left at point to send Kira Chathli packing for six and leave Surrey 35-3 in the sixth over.
Wyatt-Hodge brought up her 63rd T20 half-century – and eighth on this ground – from 30 balls, but Hawks put themselves in pole position as Grace Harris holed out to Knott at mid-on for 17 and Mary Taylor picked up her second wicket of the over as Wyatt-Hodge skewed a slower ball to Linsey Smith to depart for 65 from 40 balls.
Surrey needed 50 from the final 38 balls and 20 from the last 14, and were helped in the penultimate over when Scholfield smashed Linsey Smith down the ground for another boundary, before driving straight back to the bowler who spurned a simple catch.
Surrey needed 11 from the final over bowled by Adams and eight from the final four balls but Davidson-Richard (23*) slashed through the covers to the rope.
Two were needed from the final ball which Adams delivered outside off stump for Scholfield (30*) to carve away to the deep-point boundary to win it.
