Weekes is expected to be the Devils' primary goalie in place of Brodeur, who will be sidelined until at least February after tearing a biceps tendon in his left elbow last weekend.
Patrik Elias and Jamie Langenbrunner scored in the shootout for New Jersey, which won for the second time in six games. The Devils also got goals from David Clarkson, Elias and Jay Pandolfo in regulation.
Jokinen scored twice for Tampa Bay and Martin St. Louis tied it late in the third period, but the Lightning's three-game winning streak ended. Jokinen gave the Lightning a 1-0 lead with 3:32 remaining in the first period.
Evgeny Artyukhin set up the goal as he wheeled into the Devils zone, unloading a backhander on Weekes who blocked the shot. Mark Recchi fished out the rebound from the side of the net, feeding Jokinen who was alone in the slot.
Lecavalier had two close-in swipes in the second period, and Weekes made a left pad stop on the first. Lecavalier banked the rebound off the post.
Moments later, Mike Smith snared a sharp-angle shot from Elias.
The Devils, beaten 2-0 by Buffalo on Monday in their first game following Brodeur's injury, ended a lengthy scoring drought with goals by Clarkson and Elias late in the second period.
Clarkson picked up the puck in the Devils end and skated up the right wing boards before flipping a shot past Smith at 16:05 that appeared to glance off defenseman Lukas Krajicek's stick.
Elias put the Devils up 2-1 with a power-play goal with 1:16 remaining. Dainius Zubrus, who also assisted on Clarkson's goal, sent a backhand pass to Elias. He then unloaded a hard blast from the right circle.
Early in the third, Travis Zajac made a bold bid for a third Devils goal, but his missed with a backhander while New Jersey was short-handed.
That proved costly when Jokinen tied it at 2 with a power-play goal at 10:52. Jokinen, in the right circle, one-timed a feed from Andrej Meszaros for his fourth goal in four games.
Pandolfo gave the Devils a short-lived lead at 15:10, knocking in the rebound of Elias' shot when Smith flopped to the ice but couldn't control the rebound.
St. Louis tied it, putting in the rebound from Lecavalier's shot for a power-play goal with 3:20 remaining.
Both teams had four shots in overtime. Smith made 35 saves overall, and Weekes had 24.
Notes: Tampa Bay started a five-game road trip that will continue in New York against the Rangers on Thursday. ... Brodeur joined the Devils' lengthy list of sidelined players that includes C Brian Rolston (ankle sprain), C Bobby Holik (broken pinkie), D Paul Martin (upper body soreness) and D Andy Greene (broken hand). ... RW Radim Vrbata (groin) missed a fifth straight game for Tampa Bay with a groin injury.