Priskin scored a fine goal 12 minutes from time to give Ipswich the advantage going into the second leg - and it was no more than the Championship side deserved.
The Tractors boys had been thrashed 7-0 at Chelsea in their FA Cup third-round tie on Saturday after axing Roy Keane but there was never any danger of a repeat as, under the guidance of Ian McParland, with new boss Paul Jewell watching from the stands, Priskin gave the Suffolk side a slender advantage to defend at the Emirates Stadium in a fortnight's time.
Arsene Wenger had warned against a repeat performance of Saturday's 1-1 FA Cup draw against Leeds and named the likes of Cesc Fabregas, Theo Walcott and Jack Wilshere in his side.
However, the Gunners again failed to hit top gear and were fortunate not to lose by a bigger margin.
Both teams made positive starts but it was the hosts that created the first real chance came after 14 minutes, Priskin cutting inside from the left and unleashing a 20-yard curling effort which flashed just wide.
Arsenal continued to probe, with Walcott drilling his shot across the six-yard box but just like on Saturday, the Premier League side lacked a cutting edge in the final third and Ipswich soon gained control of the tie.
Wojciech Szczesny flapped at a ball into the six-yard box as Gareth McAuley headed over and shortly after Ipswich captain David Norris was inches away from connecting with Connor Wickham's left-wing cross.
The home fans were on their feet when Priskin netted an acrobatic overhead kick after 42 minutes, but celebrations were immediately cut short by an offside flag.
Walcott skipped into the area at the start of the second half, before tripping over the ball - which drew ironic cheers from the home fans after the Arsenal forward had admitted to diving against Leeds - before Andrey Arshavin fired an angled volley high and wide after being played in by Fabregas down the left.
Ipswich, though, refused to lay down and continued to take the game to their illustrious visitors, Wickham again catching the eye with some powerful runs.
Mark Kennedy robbed Fabregas and, with Szczesny off his line, tried an audacious 35-yard lob, but the 6ft 5ins Pole recovered quickly to make a comfortable save.
Emmanuel Eboue was then caught in possession by Priskin, but recovered to block the Ipswich striker's goalbound shot.
Just after the hour, somehow Arsenal survived when a long ball caught out Koscielny.
For a moment Priskin was clean through on goal, however Johan Djourou made a brilliant interception to knock the ball off the striker's toes from inside the six-yard box.
As the rain lashed down on Portman Road, Arsenal continued to give the ball away cheaply although Gibbs fired a shot wide from 20 yards.Then, after 67 minutes, Wenger made a change when Alex Song replaced Wilshere and Marouane Chamakh came on for the ineffective Nicklas Bendtner.
Arsenal immediately looked to raise the tempo, Marton Fulop making a smart save at his near post after Walcott had been played in by Fabregas. The Arsenal skipper then put Gibbs' cross over from four yards out off his thigh after Chamakh had just failed to connect.
Ipswich made the breakthrough after 78 minutes, though, when Colin Healy's through-ball split the Arsenal centre-backs and Priskin got ahead of Djourou to calmly slot into the bottom right corner.
Wenger immediately sent on pacy Mexican forward Carlos Vela, in place of Arshavin, and Fulop had to block from Walcott but Szczesny was forced into another save from Carlos Edwards late on as the Tractor Boys closed out for a famous victory.