Ayite opened the scoring for the Cottagers in the 17th minute before Scott Malone doubled their advantage on the half-hour mark, and Stefan Johansen made the game safe in the second half.
Christophe Berra's injury-time header proved to be only scant consolation for Ipswich.
Manager Slavisa Jokanovic responded to his side losing 4-2 at Derby by replacing much criticised keeper David Button for Marcus Bettinelli, while Tim Ream lost out to Michael Madl.
For Ipswich, Mick McCarthy made one replacement from the side that beat Wigan 3-0 with Tommy Smith replacing the injured Steven Taylor.
The home side started the game on the front foot, with an intent to attack, and they took the lead after 17 minutes.
Ipswich goalkeeper Bartosz Bialkowski was forced into a fine save from a low Chris Martin shot, before Ayite netted his third goal in four days from the rebound.
Ipswich struggled to get into the game, their first effort being a 28th minute shot from Freddie Sears which hit the side netting after a break from the halfway line.
Fulham doubled the lead two minutes later, when Malone was left unchallenged in the area to curl his effort home past Bialkowski.
The home side were still looking for a third strike, and Bialkowski had to be equal to a 39th minute curler from the edge of the box by Kevin McDonald.
David McGoldrick put his effort wide on the stroke of half time for the visitors, who looked to toil in the Craven Cottage sunshine.
After his recovery from a groin injury, Tom Lawrence was put on in the second half for Ipswich, with Tommy Smith making way.
However, it was Fulham who had the first effort of note in the second half, when a free kick by Ayite on 50 minutes rattled the crossbar.
Bettinelli was forced into action on 53 minutes with a fantastic point blank save with his foot, after Sears collected a rolled pass from McGoldrick.
The game was effectively over on the hour mark, although not before the Fulham goalkeeper had to keep out a McGoldrick drive, when Johansen found space in the box, before turning inside and slotting home.
Bialowski had to hold another curling effort from Ayite on 65 minutes, although the game was to settle into a pattern where the hosts were more than comfortable.
Ayite was substituted to a rousing reception on 86 minutes, with Neesens Kebano replacing him.
However, despite a Berra header in injury time for Ipswich, the west London outfit ran out deserved winners - and kept their play-off hopes alive.
Additional reporting by the Press Association.