Midfielder Diame – brought to the KC Stadium by Steve Bruce in August – rose well to head Andrew Robertson’s pinpoint cross beyond Julian Speroni to put the Tigers ahead.
With time almost up, Jelavic ran onto a defence-splitting pass from Jake Livermore and set himself before slotting the ball beyond the Palace keeper.
City were good value for the win – their first at the KC Stadium this season – which sees them leapfrog their visitors who created little throughout.
Jelavic and Abel Hernandez both missed decent first half openings before Diame rose unchallenged on the hour mark to give City the lead.
Neil Warnock threw on Marouane Chamakh and Dwight Gayle soon after falling behind in a bid to get something from the game but the home side continued to look comfortable with the visitors struggling to unlock the home defence.
And the game was made safe with just a minute remaining thanks to Jelavic’s cool finish.
Palace thought they had managed to get a late goal back within moments of the Croatian international’s goal but Liam Rosenior was in the right place to hack Damien Delaney’s header off the line.
Just as pleasing to Hull boss Steve Bruce will be his side’s clean sheet, also a first since the opening day of the season.
Bruce’s side had shipped 10 goals in their previous four league matches, and another three in the Capital One Cup loss at West Brom, prompting a return to last season’s preferred 3-5-2.
That meant James Chester was back at the heart of defence and Hull were certainly a tighter unit for the tactical switch.
It also allowed wing-backs Ahmed Elmohamady and Robertson greater freedom to attack and their energy on the flanks was a constant thorn in Palace’s side.
The away side, having registered back-to-back wins, came back to earth with a bump and had to wait 74 minutes for their first shot on goal.
They started brightly enough, Jason Puncheon and Yannick Bolasie both worrying Hull with their direct running but the back three largely stood firm, allowing Elmohamady and man of the match Robertson to pour forward.
Palace held out for the break, though not before Diame and Hernandez had spurned decent openings.
Palace began the second half with greater intent but found themselves forced back as Hull responded in kind.
First Jelavic drilled a free-kick into the wall after more good work by Robertson; then Elmohamady bustled past two men on the right and pulled the ball back to the edge of the area.
Unfortunately for City Curtis Davies was the man arriving and the defender cleared the crossbar by 10 yards.
But the pressure finally told with half an hour to go. Huddlestone and Hernandez had driven Hull forward and when the ball broke for Robertson he curled a wonderful cross into the six-yard box. Diame obliged with a crisp header, continuing his rich vein of form.
Neil Warnock’s side were forced into action and finally made 39-year-old goalkeeper Steve Harper work on his return in place of the injured Allan McGregor.
Bolasie was the man to do it, unleashing a powerful strike from 30 yards and forcing a smart reaction save as Harper palmed over.
But by committing men forward Palace were increasingly open at the back and Livermore sliced them in two with a superb through ball for Jelavic in the 89th minute.
The Croatia striker calmly evaded Martin Kelly then deposited his shot into the far corner to seal the points.
Charlie Nicholas’ verdict
“Going back to a back three is definitely a positive for Hull. It gives them security at the back and allows Elmohamady and Robertson to blast forward. One thing about the pair of them is they put in lovely service. They are excellent crossers and good passers. Diame did well and I think it suits him sitting in behind Huddlestone, who has a very nice attitude on the ball and ability on both feet.
“The bad news for Palace was Scott Dann got a bad injury. He went for a block, but it looks as if he twisted his ankle. It probably leapt into his knee and he had a brace on, so I hope there is better news than it looked. It looks as though he may be out for a while.”