In the seventh round of the tiebreaker, Svechnikov went to his backhand and slid the puck between the pads of Ottawa goalie Craig Anderson.
Tomas Tatar had a goal and an assist, and Frans Nielsen, Gustav Nyquist and Dylan Larkin scored in regulation for Detroit, Nick Jensen had two assists and Petr Mrazek stopped 41 shots.
Kyle Turris and Fredrik Claesson scored 4:09 apart in the third period to tie the score for the Senators, who pulled into a tie with Boston for third place in the Atlantic Division. Ottawa is technically ahead at the moment with a game in hand on the Bruins.
Erik Karlsson had a goal and an assist, Alexandre Burows also scored and Turris also had two assists for the Senators.
After a scoreless opening period, the goals came quickly in the second. Nielsen tried to pass the puck to teammate Riley Sheahan from behind the Ottawa net but it hit the skate of Senators defenseman Ben Harper and bounced through Anderson's legs to give Nielsen his 16th of the season at the 40-second mark.
The Senators tied it 1:46 as Burrows' seemingly harmless wrist shot bounced off the top of Mrazek's glove and into the net.
Detroit regained the lead short-handed at 6:53 as Larkin was skating the puck past the Ottawa net and threw a backhander that found a way in between Anderson and the post.
Yet another bizarre tally pulled Ottawa even on the power play just 56 seconds into the third. Karlsson, back in the lineup after missing two games with a left foot injury, skated below the Detroit goal line and lifted the puck to the front of the net from the corner. It hit the back of Mrazek's stick as he dropped to play it and skidded in through his legs.
The Wings struck on a power play of their own at 5:53 as Nyquist lifted the puck over a diving Anderson, who came up short in his attempt at a poke check.
Detroit added another power play goal at the eight-minute mark to take a 4-2 lead. Mike Green's point shot deflected off the blade of Nyquist's stick and fluttered past Anderson.
Turris snapping a wrist shot past Mrazek on the stick side at 8:22 to pull Ottawa within one, and Claesson's point shot beat Mrazek on the glove side with 7:29 remaining.
NOTES: Senators C Colin White also made his NHL debut. White, who signed a three-year entry-level deal on Sunday, won a gold medal with Team USA at this season's world junior championship. ... Red Wings LW Justin Abdelkader (United States) and both of the teams goaltenders, Jimmy Howard (United States) and Mrazek (Czech Republic), confirmed that they will play in the world championship. Red Wings equipment manager Paul Boyer will serve in that role for Canada at the worlds.
Senators: Host to Detroit on Tuesday.
Red Wings: At Ottawa on Tuesday.