It was the Oilers' first win against a team above the playoff cutline in a calendar month.
Tampa Bay started the scoring with a power-play goal less than three minutes into the game as Matthew Carle made a cross-ice pass to Kucherov at the left face-off dot. Kucherov beat Oilers starter Ilya Bryzgalov with a quick shot glove-side.
Edmonton responded a minute-and-a-half later, however, as a stick got caught in the skates of Martin St. Louis in front of the Tampa net before Eberle could fish it out and lift his 13th of the season past Lightning goalie Ben Bishop.
Bishop left the game after the play with what originally looked like an equipment problem but ended up being an injury to his right hand that prevented him from returning and back-up Anders Lindback came in as his replacement, just as Alex Killorn took a double minor for high-sticking.
Edmonton went up 2-1 on the first half of the man advantage on their second shot of the game, just past the six minute mark, as an Eberle shot pinballed in front and was inadvertently deflected past Lindback by defender Sami Salo. It was Eberle's second goal in less than two minutes after only scoring one in his previous 11 games.
The Oilers took a two-goal lead with five minutes left in the first period as a Tampa giveaway behind their net eventually allowed Gazdic to tip his second career goal past Lindback off a point shot from Justin Schultz.
The Lightning started to change the direction of the game in the second period and were rewarded for it just under nine minutes in as a stolen puck at the Oilers blue-line eventually led to a long screened-shot goal by Hedman.
Tampa Bay tied it 3-3 with 5:35 remaining in the second period as Filppula hooked off the half boards and picked the bottom corner of the net with a quick wrist shot for this 16th goal of the season.
Lindback kept the Oilers from regaining any momentum on a late second period power play, making big stops on Eberle and Gordon to keep it 3-3 after 40 minutes.
Both teams hit posts early in the third period, including one by Edmonton's Ales Hemsky that prompted the goal judge to turn on the red light before a video review determined it didn't go in.
Edmonton moved back in front eight minutes into the third period as Gordon did a good job to fight off a defender and swat home the rebound from his own backhand shot in tight for his second goal in as many games.
The Oilers went up 5-3 just past the mid-mark with the teams playing four aside as Hall made a perfect redirection of an Andrew Ference point shot for his 16th goal and team-leading 41st point of the season.
Both teams return to the ice on Tuesday as the Lightning finish a four-game trip in Winnipeg and the Oilers play host to the St. Louis Blues.
Notes: It was the second and final meeting between the two teams this season. The Lightning defeated the Oilers 4-2 on Nov. 7 in Tampa Bay… Lightning forward Steven Stamkos (broken right tibia) has begun skating in full gear but is still not ready to return to the team. However, the Lightning have persevered without their star centre, coming into the game with a 13-7-4 record while he has been out… Tampa goalie Bishop could be a big loss if he is out for long after his first period injury. Bishop earned his league lead-tying fourth shutout of the season in a 2-0 win over Calgary on Friday and entered the contest against the Oilers having earned at least a point in 10 straight games (8-0-2) and allowing two or fewer goals in eight consecutive starts and 26 of 31 games… Bolts defenceman Sami Salo was back in the lineup after sitting out the last four games due to an upper-body injury… Oiler Eberle returned after missing Edmonton's last game against Anaheim with a knee injury. Defencemen Corey Potter (groin) and Philip Larsen (illness) are on injured reserve for Edmonton.