Wing Senior went over twice in the first half after team-mate Josh Charnley's effort had cancelled out Ryan Hall's opener for Leeds, for whom Brodie Croft also touched down before the break.
In balmy, energy-sapping conditions after the hottest day in the UK this year, Leigh scored a fourth try through centre Umyla Hanley early in the second period and then withstood heavy pressure to seal a deserved win.
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Ryan Hall thought his late effort for Leeds had tied the scores, but his foot was clearly in touch as he grounded the ball in the corner and Leigh clung on.
Hall touched down in the corner for the opening try in the second minute after taking Jake Connor's quick pass following a high kick, but Leigh's response was immediate as Charnley raced over following Hanley's line break.
Connor's penalty in front of the posts for a shoulder barge on Kallum Watkins edged Leeds 8-6 ahead, but Leigh piled on the pressure midway through the first half and got their reward.
Senior powered over near the touchline and despite Leeds full-back Lachlan Miller's brilliant tackle, the video referee ruled the Leigh winger had dotted down before going out of bounds to put his side 10-8 up.
Leeds came roaring back when Croft burst through a gap under the posts for a converted score to nudge them 14-10 in front.
Rhinos centre Jack Bird went off for a head injury assessment following a clattering tackle and was replaced by teenager Zak Lloyd, whose terrific last-ditch tackle halted Charnley soon after.
But quick hands saw Leigh spin the ball left for Senior to go over on the outside to level the scores at 14-14 at half-time after Cook had missed his second conversion attempt.
Connor landed his second penalty to give Leeds a two-point advantage soon after the restart following Cook's off-the-ball block on Ash Handley.
Leigh responded again though, carving open the Rhinos' defence via another slick handling move that sent Hanley over for the visitors' fourth try of the match, with Cook adding the extras to put them 20-16 in front.
The visitors were reduced to 12 men when second row Andy Badrock was sin-binned for his dangerous tackle on Harry Newman with 17 minutes to go.
Leeds threatened to make their extra man count, but they lacked composure and failed to execute in the face of some desperate, scrambling defending from Leigh.
With three minutes remaining, Leeds thought they had drawn level when Handley sent Hall over in the corner following Connor's high kick.
But the video referee revealed the winger's foot was in touch as he grounded the ball and Leigh held firm before celebrating an impressive win.
Leeds' defeat gives Wigan Warriors the chance to move level with them at the top of the table on 36 points when they visit struggling Bradford Bulls on Sunday.
Super League 2026 Round 21 Schedule and Results - live on Sky Sports
Thursday August 13
Hull KR 28-10 Catalans Dragons
Leeds Rhinos 16-20 Leigh Leopards
Warrington Wolves 30-10 York Knights
Friday August 14
8pm: Hull FC vs Castleford Tigers
Saturday August 15
5.30pm: Wakefield Trinity vs St Helens
6.30pm: Bradford Bulls vs Wigan Warriors
8pm: Tolouse Olympique vs Huddersfield Giants
Sky Sports will show every game of the Super League live this season, including two matches each round exclusively live. The remaining five matches will be broadcast on Sky Sports+.
