Premier League: Cardiff 0 Hull 4
The hosts went into the game desperate for just a second league triumph since Ole Gunnar Solskjaer took the reins.
Things did not go to plan, however, as Hull - buoyed by having big-money January recruits Jelavic and Shane Long back in the fold after they were ineligible for the FA Cup 1-1 draw with Brighton and Hove Albion - grabbed only their second Premier League victory in eight games.
Having been set on their way by Tom Huddlestone's deflected strike, the visitors took a fortuitous 2-0 lead into the break as Jelavic netted his second goal since signing in the transfer window following a one-two with Long.
Things did not improve for Cardiff after the break as Jelavic headed home his second before Jake Livermore stroked home a fourth.
The misery ended there for the hosts, while Hull could take plenty of encouragement from a triumph that leaves them looking up rather than over their shoulders following a recent poor run.
Boasting plenty of pace on the counter-attack, Cardiff looked threatening during the early stages - even if a series of promising openings ultimately amounted to little.
They were almost rewarded for their industry in the 12th minute, though, following some beautiful skill from Wilfried Zaha.
The on-loan winger, who had struck straight at the goalkeeper from just outside the area a few moments earlier, jinked his way past three defenders in the box and forced a smart save from Allan McGregor.
The Scotland international had to be alert again soon after when denying Kenwyne Jones.
Not to be outdone, McGregor's opposite number, David Marshall, matched his countryman by pulling off a fine stop of his own as he kept out Jelavic's effort from point-blank range following the striker's fine one-two with Long.
Yet Marshall could do little as Hull took the lead from the resulting clearance. The ball found its way to Huddlestone, who unleashed a vicious strike that deflected past the shot-stopper.
To their credit, Cardiff responded impressively and almost levelled matters when Jones headed just wide from three yards.
But Hull hit their hosts with a sucker-punch as the Jelavic-Long combination bore fruit once more in the 38th minute.
Having got on the end of his partner's throughball, Long - who may have been marginally offside - returned the favour and Jelavic slid the ball past Marshall emphatically.
Whatever Solskjaer said to his players at half-time did not appear to work as Jelavic piled on further misery.
Having struck wide with a decent chance in the immediate aftermath of the restart, he made no mistake when glancing Liam Rosenior's cross home.
And their dominance was further illustrated in the 67th minute as Livermore side-footed past Marshall after being found by Ahmed Elmohamady.
Hull boss Steve Bruce was subsequently afforded the luxury of being able to withdraw the hard-working Long as the visitors eased to the points.