The Senators jumped out to a 4-1 lead before squandering it, then ran away with the contest with three third-period goals.
Kovalev scored his first goals in 11 games and his first since recording a hat trick in a Dec. 12 win over the Carolina Hurricanes at home.
The previous three-goal effort provided his only goals in the 27 games he'd played, not including three he missed, since scoring on Oct. 29.
Against the Flyers, though, the Russian ran riot before a crowd of 17,153 at Scotiabank Place and recorded his first career four-goal contest. It was his second hat trick of the season and 12th of his career.
His first goal came against Flyers starter Michael Leighton, who was pulled in favor of Brian Boucher after giving up two goals on five shots.
Senators starter Pascal Leclaire didn't fare much better, allowing three goals on 10 shots before making way midway through the game for Brian Elliott.
Chris Kelly had his first career two-goal game for the Senators (22-16-4), who won for the second straight game and fourth time in the past five contests.
Jarkko Ruutu added a goal and an assist for Ottawa, and Ryan Shannon and Anton Volchenkov had three helpers apiece.
Elliott finished with 10 saves to record the win.
The Flyers had been hot recently, going 4-0-1 in the past five games with their only loss coming on New Year's Day in overtime to the Boston Bruins at the Winter Classic.
Sunday's game was no classic, however, after they rallied to get back into the contest in the second period, only to fall apart in the third.
Arron Asham scored once and set up another goal, and Ian Laperriere, Darroll Powe and Mike Richards also added goals on the power play.
James van Riemsdyk chipped in with a pair of assists and Boucher made 21 saves.
It was not a good day to be a goaltender in Ottawa.
Laperriere gave the Flyers a 1-0 lead at the 2:10 mark of the first period when he beat Leclaire, making his sixth straight start since returning from a fractured cheekbone, from the right wing just a step inside the blue line. It was Philadelphia's second shot of the game.
Leighton was also making his sixth straight start and had been unbeaten at 4-0-1 in the previous five games with a 1.56 goals-against average and .947 save percentage.
Those numbers took a hit once Kovalev had the Senators on level terms 2 1/2 minutes after Laperriere's goal.
At the 9:07 mark, Kelly beat Leighton with a high shot to the short side and Flyers coach Peter Laviolette had seen enough, giving the goaltender the hook in favor of Boucher.
His first contribution was to scoop the puck out of his net after just 80 seconds when Chris Neil fed Ruutu in front to make it 3-1 and Kovalev added another on a rebound after Boucher made his first save of the game.
In 10 shots on goal, Ottawa had scored four times.
Powe cut the lead to two early in the second period on another goal from close range that Leclaire should have stopped, so when Asham made it 4-3 at the 5:53 mark of the second, Senators coach Cory Clouston replaced Leclaire with Elliott.
By then, the Flyers had the momentum and Richards tied it with a bullet to the top corner during a 5-on-3 power play.
However, Kovalev broke his long goal drought early in the third after Boucher couldn't control a shot by Shannon.
Kelly added another with the teams playing four players a side and Kovalev scored his fourth on a shot from the slot.
It's the fifth time in franchise history a member of the Senators has scored four goals.
Notes
The Senators made defenseman Alexandre Picard, a former Flyer, a healthy scratch for the second straight game. … The teams split the first two games of the season series in Philadelphia earlier this year. … Philadelphia goalie Ray Emery is still recovering from abdominal surgery last month and missed a chance to play in Ottawa for the first time since leaving the Senators.