Ottawa has been eliminated in the playoffs each of the last three seasons by the Toronto Maple Leafs while Joseph usually played the starring role.
In his first appearance against the Senators for his new team, Joseph surrendered five goals on 34 shots. He fell to 13-14-1 lifetime against Ottawa in the regular season.
"We came in here at a bad time," Joseph said. "Ottawa is too good to ever be slumping. I could've played better at my end. Overall, it wasn't our best game."
"I don't know why we're successful when we play against Joseph," said Senators winger Magnus Arvedson, who scored in the first period. "It seems as though all the games are big, whether it's in Toronto or Stanley Cup champion Detroit. It shouldn't be like that, but it is."
Smyth scored 45 seconds into the second period to snap a 1-1 tie and Van Allen tallied just over 13 minutes later for the Senators, who stopped a four-game winless streak. Ottawa also snapped a four-game skid against Detroit.
"We definitely thought coming in that this would be a test for our hockey club," Senators coach Jacques Martin said. "Thanks to to key saves by Patrick (Lalime) and contributions from different people, we were able to win."
Brett Hull scored on the power play and rookie Henrik Zetterberg registered a shorthanded goal for Detroit.
Smyth made a nice individual effort pay off on the go-ahead goal early in the second when he moved around defenseman Jiri Fischer at the top of the right faceoff circle and into the slot, where he beat Joseph with a backhander.
Van Allen provided a 3-1 lead with 6:03 left in the period, taking a pass from Petr Schastlivy and beating Joseph with a wrist shot.
With defenseman Maxim Kuznetsov in the penalty box for cross-checking, the Red Wings tallied while shorthanded with 9:51 left in the third. Zetterberg redirected Hull's shot by Lalime for his third goal, pulling Detroit within 3-2.
It was the Red Wings' league-leading sixth shorthanded goal in 11 games.
Still on the power play, the Senators struck just 49 seconds later when Schastlivy put a backhander off the stick of Red Wings left wing Kirk Maltby and past Joseph for his fourth goal and a 4-2 advantage.
"The second half of the game was not as strong as we would have liked," Red Wings coach Dave Lewis admitted. "You want to get into a rhythm, but every game is different. We got a shorthanded goal to get within one, which usually affects your opponent, but we didn't get out of the penalty-kill without them scoring."
Two minutes later, Todd White scored his first goal of the season to give the Senators the final margin.
Lalime, who lost his two previous decisions against the Red Wings, made 20 saves.
Lalime's night did get off to an auspicious start as Hull fired a one-timer from left circle by him 5:41 into the contest. It was Hull's third goal and the Red Wings' 13th on the power play this season. Detroit has scored with the extra attacker in each game.
Just 55 seconds after Hull's tally, Arvedson tied it with his first goal of the season.