The hosts came close to breaking the deadlock in the opening five minutes when Jesse Joronen tipped Christophe Berra's header over the bar, before Luke Hyam dragged a shot wide from the resulting corner.
Chris David had Fulham's first effort on goal minutes later, but after receiving the ball in a promising position outside the box, he could only direct his low drive wide of the post.
Ipswich were the next to threaten and Cole Skuse nearly found the net with a dipping volley that went just over the bar.
In the 32nd minute, Ipswich made the breakthrough as Daryl Murphy raced onto Hyam's through ball before drilling an effort past Joronen at the keeper's near post.
Shortly after the break, Kostas Stafylidis came close for the visitors, but his fierce effort flew just over the bar.
Just past the hour-mark, the hosts grabbed their second of the match as David McGoldrick fired home a loose ball from the edge of the box.
With four minutes of the game left to play, Fulham pulled a goal back when Tim Hoogland cut inside from the right, before drilling a deflected strike that ended up in the back of the net.
Fulham pushed hard for an equaliser in the closing minutes, with Dembele heading just over the bar from a corner, but Ipswich held on to begin their new season by claiming all three points.