Pisani was able to get a handle on the puck during a mad scramble in front of the Calgary net and slip it low glove side on Flames netminder Miikka Kiprusoff.
It was the third straight game where the Oilers overcame a two-goal deficit to pull out a victory.
Calgary's top line of Jarome Iginla, Daymond Langkow and Kristian Huselius cycled the puck well down low in the opening minutes of the game but couldn't generate a good scoring chance.
On their next shift, Iginla's line picked up where it left off, this time with better results. After battling to keep the puck in the zone, Huselius wristed a shot from the left faceoff circle that beat Morrison five hole.
After an impressive penalty kill, the Oilers scored on their first shot of the game, a blast by Igor Ulanov from the high slot.
Three minutes later, Calgary came back down the ice to score their second of the game. Iginla carried the puck over the Oilers line then fed Langkow who rifled one top corner blocker side behind Morrison.
The Flames almost made it 3-1 while shorthanded when a Morrison miscue caused him to cough up the puck to Shean Donovan, but the Oilers goaltender was able to get back in position to stop the shot.
Late in the first, Shawn Horcoff
scored to tie the game at 2-2. Horcoff took a pass in the slot and wristed it on net. It was tipped in front by a Flames defender and through the legs of Kiprusoff.
Shortly after the Oilers goal, penalties to Matthew Lombardi and Bryan Marchment sent the Oilers on a two-minute two-man advantage. Although Edmonton was able to pin Calgary in the zone for the duration, they were unable to score.
Right at the end of the period, Kiprusoff came up huge for the Flames after Shawn Horcoff
was denied on a glorious shorthanded opportunity.
Although the Flames had a 6-0 edge in shots to start the game, Edmonton led Calgary 10-9 after one with the game knotted at 2-2.
The Oilers had a pair of great scoring chances early in the second. First, Kiprusoff stopped a tip-in attempt by Shawn Horcoff
after a nice three-way passing play that also involved Smyth and Hemsky.
On the same shift, Hemsky pulled a nifty move to get around the Flames defender before attempting a wrap-around. Kiprusoff was barely able to come across and keep the puck out.
Several minutes later, Dion Phaneuf scored on a blast from the point that beat Morrison five-hole. The goal was Calgary's third on 10 shots, chasing Morrison from the nets in favour of Jussi Markkanen for the second straight game.
Markkanen didn't fare much better at first, getting beat by Steve Reinprecht on a Robyn Regehr tipped shot 31 seconds after the Phaneuf marker.
Late in the period while on the power play, Edmonton was able to muster some pressure on the Flames. Kiprusoff had to come up big on several occasions as the Oilers pressed to get back within a goal.
Just as the penalty to Jason Wiemer was about to expire, the Oilers scored to make it a 4-3 game. Igor Ulanov potted his second of the game with a screened blast from the point.
After two periods, Edmonton had a slight 17-14 shot advantage.
Markkanen made a big stop early in the third period to keep Edmonton in the game. Chuck Kobasew and Matthew Lombardi broke in on a two-on-one which ended in Kobasew tipping a Lombardi feed on net. Markkanen was able to come across to stack the pads.
Midway through the period, Phaneuf and Todd Harvey exchanged words before pushing and shoving their way to coincidental roughing minors.
Daymond Langkow rang a shot off the iron about a minute later as the teams started to trade chances.
The Oilers netted the tying goal on Horcoff's second of the night. The centreman fired a desperation shot from an impossible angle that found its way off Kiprusoff's pads and in.
Mike Morrison was named the starter in goal tonight for the Oilers, Miikka Kiprusoff for the Flames... Stephane Yelle (ankle) and Chris Simon (healthy) both did not suit up for Calgary... Ales Hemsky
had his ninth multi-point game of the season, tying Jarret Stoll for the team lead... Horcoff netted his seventh multi-point outing... Three of Calgary's four goals were scored by Edmonton natives: Phaneuf, Reinprecht and Langkow... Ulanov's two goals was a career high in one game... Moreau now has points in seven consecutive games to extend his career high streak.