Briers, the most prolific drop-goal kicker in the history of Super League, had missed with two short-range efforts to break the deadlock in the last five minutes but succeeded at the third attempt to break the hearts of the injury-hit champions.
The Rhinos looked to have ended Warrington's five-match winning run after outscoring their hosts by four tries to three but Zak Hardaker, kicking in the absence of Kevin Sinfield, was able to land only one of his four conversion attempts.
England winger Ryan Hall scored two tries, one of them a 100-metre solo effort, and 18-year-old Liam Sutcliffe, deputising at stand-off for the injured Sinfield, celebrated his new four-year contract with his sixth try in eight Super League appearances but it was not enough to hold off Warrington's late onslaught.
Leeds led 8-6 at half-time after taking their chances as Warrington squandered a host of opportunities. Man of the match Micky Higham put the home side in front after five minutes when he followed up his own delicate grubber kick to touch down and Brett Hodgson kicked the first of his three goals.
The Rhinos were always dangerous from long range and they duly took the lead with the aid of two tries in five minutes. Hall scored the first, with the aid of a deft one-handed flick pass from his centre Joel Moon. Young Sutcliffe then went 80 metres to score the second after pouncing on a stray pass from Stefan Ratchford.
The champions extended their lead two minutes into the second half when Hall intercepted Simon Grix's pass meant for Joel Monaghan to sprint the length of the field to score his second try.
Hardaker was once more off target with the goal and the Wolves drew level on 54 minutes when Ratchford spotted a gap in the Leeds defence and squeezed over from dummy half for his side's second try.
The momentum was clearly with Warrington at that stage but Leeds struck once more on the hour when Hardaker took a return pass from Sutcliffe to go in for a fourth try and this time he added the goal from close to the posts.
But the scores were level once more when Monaghan palmed the ball back from Briers' high kick for Hodgson to touch down. Hodgson's third goal made it 18-18 and, in a frantic finale, Briers was twice wide with drop-goal attempts before finally hitting the target.
Wigan fought back with four late tries to claim a gutsy last-minute 26-20 win against Bradford in their clash at the DW Stadium.
The Warriors got off to a perfect start when Josh Charnley crossed in the second minute, but that was as good as it got for the home side in yet another poor opening 40 minutes from the league leaders until a second-half fightback and 80th-minute score for Chris Tuson saw them home.
Bradford responded with tries from Michael Platt, Adam Sidlow and John Bateman, and Jamie Foster added a penalty to stretch the advantage further. However, Wigan struck back as Charnley crossed again with Tuson and Iain Thornley also making it over. Tuson's second converted try - with Pat Richards slotting his third goal - sealed victory.
The Warriors hit the front thanks in great part to Ryan Hampshire, who was standing in for full-back Sam Tomkins, as he steamed through a gap before sending Thornley clear who linked up with Charnley for the opening try.
Richards missed the conversion and the Bulls were not behind for long. Foster found an overlap down the left and sent Sam Wood racing down the touchline. He drew in Hampshire to allow Platt to touch down and Foster tagged on the extras for a 6-4 lead.
Bradford had the upper hand and when Hampshire failed to hold a Luke Gale bomb, Sidlow was on hand to score, Foster converting. Bateman then saw off the attention of three Wigan defenders to touch down as Bradford cruised into half-time.
After the break, Foster extended their lead for an 18-4 advantage. Wigan finally broke the visitors down in the 64th minute when Hampshire again popped up on the right and he sent in Charnley for his second try of the night. Richards was off target and they now trailed by 10.
The Warriors were showing more urgency now and they made it back-to-back scores as Liam Farrell broke clean up the middle before sending out a dubious ball for Tuson to race away and score under the posts. Richards made no mistake this time as they trailed 18-14 with 15 minutes to go.
However, a knock-on deep in their own half looked like undoing all of their hard work and Michael McIlorum was penalised for a spear tackle, Foster slotting over another penalty to push them six points in front.
The Warriors were over the whitewash again through Thornley and Richards calmly slotted the extras to tie the game at 20-20. And Tuson kept his composure despite losing his footing before scoring the winning try in the final minute, with Richards converting.