Sidney Crosby, Chris Kunitz and Pascal Dupuis also scored for Pittsburgh, and Tomas Vokoun made 30 saves. Down 2-1 after the first period, the Penguins scored twice in a 40-second span early in the second and added four goals in the first 10 minutes in the third.
Game 5 is Friday night in Pittsburgh.
The Senators, coming off a double-overtime victory Sunday night in Game 3, opened the scoring on Michalek's short-handed goal at 2:29 of the first period. With defenseman Sergei Gonchar in the penalty box, Alfredsson fed the puck up the middle to a streaking Michalek, who broke through the defense and beat Vokoun low on the glove side for Ottawa's second short-handed goal of the series.
It was the Senators' first lead of the series. It wouldn't last.
Neal tied it with 5:04 left in the period, picking up the loose puck in the slot and putting it in the back of the net. Turris put Ottawa back in front off a rebound with 3:45 left in the first.
Pittsburgh tied it early in the second when Kunitz got behind the defense and went in alone to beat Anderson. Less than a minute later, Anderson gave up a rebound on Kris Letang's shot that landed on the tape of Iginla's stick to make it 3-2.
Neal started the third-period flurry with a power-play goal at 1:59. Dupuis added a short-handed goal at 8:07, Crosby followed at 8:39, and Iginla scored on a power play at 9:53.
Alfredsson finished the scoring with a late power-play goal for his 100th career playoff point.