McLean put them ahead, keeping his composure to poke home after good work from Robbie Brady.
Cairney made it 2-0, after catching the goalkeeper off his line with a fine left-foot finish from 25 yards.
Ipswich halved the gap after the break when substitute Jason Scotland tucked in a rebound but Stewart sealed it on the break in added time.
Hull caretaker manager Nick Barmby, who is expected to be confirmed as permanent boss on Monday, said: “We knew that they’d come up against us as they had nothing to lose at 2-0 down as we saw what happened against Barnsley when they won 5-3.
“The main thing is that we’re back to winning ways, we’ve scored three goals and I’m pleased.
“Aaron McLean was a deserved man of the match and I thought he was tremendous but I don’t think you can single out a single player.”
Barmby rotated his squad with Robbie Brady, Joe Dudgeon and Richard Garcia earning rare starts and Seyi Olofinjana and Jay Simpson on the bench,
Vito Mannone started in goal after joining on loan from Arsenal in midweek as Peter Gulacsi is out for eight weeks with a knee injury. Martin Pusic is also out.
For the full report and reaction read Sports Monday free inside your Yorkshire Post.