Posh, who lost top scorer Marcus Maddison just before the game through illness, were behind after just three minutes as two of Fleetwood's non-league captures made their mark.
Ash Hunter's pass split the defence and Sowerby, signed from North West Counties side Squires Gate, raced in with a cool finish.
Fleetwood might have extended their lead before the half hour when former Peterborough man David Ball was released down the right and Hunter was just inches away from connecting with his cross.
Ball then had a curling free-kick tipped over the crossbar by Luke McGee as Fleetwood pressed for a second.
Shaq Coulthirst - a trialist with Fleetwood last term - went closest to levelling things for Posh before the break. Leonardo Da Silva Lopes slipped the shackles of his marker before Coulthurst turned in the box but fired over the angle.
Fleetwood continued to dominate after the break but a mistake nearly proved costly when Coulthirst slipped between two defenders and fired against the palms of Chris Neal.
Fleetwood centre-back Ashley Eastham almost made pressure count at the other end as his point-blank header was blocked on the line by Ryan Tafazolli.
The defender was at fault moments later when Fleetwood finally stretched their lead.
He was too easily turned by Devante Cole, who swung a 45-yard cross from the right flank and Bobby Grant slid in at the back post to complete the move in spectacular fashion.
Cole could have had one for himself with 15 minutes remaining, sauntering on to Ball's pass before rattling the woodwork.