Hull went into the game without a league goal in 2014 but it took just 12 minutes for Shane Long to break the deadlock on his home debut as he raced through on goal and finished calmly past Hugo Lloris.
Spurs managed to retrieve a point thanks to a wonderful turn and finish from Paulinho just after the hour mark, but neither side could find a winner in an open finale.
The result leaves Tottenham two points adrift of fourth-placed Liverpool, having now played a game more, while Hull remain in the relegation scrap but have at least ended a run of four straight defeats.
Hull had started brightly, and it was former Spurs man Tom Huddlestone who played a key role in the first chance.
He caught Spurs cold by floating a wide free-kick towards Jelavic at the near post and, although the Croatian’s contact was not the cleanest, his volley nearly crept under the crossbar.
With both sides lining up 4-4-2, Roberto Soldado did well to find Emmanuel Adebayor in space at the edge of the area and his crisp first-time effort was well saved by Steve Harper.
After 12 entertaining minutes, the game got the goal it deserved as Long and Jelavic teamed up to lethal effect.
Jelavic played provider, showing good enterprise to hook the ball into space with his back to goal. Long galloped after it and immediately gained a yard on Jan Vertonghen and the sluggish Michael Dawson.
By the time the Irishman reached Lloris he had created a one-on-one opportunity and made no mistake from 12 yards.
Spurs were not immediately panicked by the goal and enjoyed plenty of the ball for the next quarter of an hour.
Adebayor, Soldado and Christian Eriksen all featured but Hull defended stoutly to keep them at bay.
Curtis Davies, in particular, was turning in a fine shift.
Spurs, though, had become a touch complacent at the back and needed Kyle Walker to make a clinical sliding challenge when Jelavic swooped on a loose crossfield pass.
The former Everton man again threatened when Dawson’s foul on Long gave Hull a free shot on goal in the 39th minute. Jelavic curled his effort a yard wide with Lloris rooted.
Spurs continued to struggle against Jelavic after the interval, Walker dithering near the corner flag and complaining to the referee as the Croatian picked his pocket.
By comparison, it was not until the 56th minute that Harper was called on again.
Vertonghen’s rampaging run got Spurs going before Paulinho’s shot was diverted into Soldado’s path. He took aim from a tight angle but Harper kept his cool to palm clear.
The equaliser felt likelier now and arrived just after the hour mark.
A short free-kick resulted in Danny Rose’s hopeful shot from 30 yards and, while that was never likely to worry Harper, it did reach Paulinho, who turned and found the roof of the net from 18 yards.
Tim Sherwood’s side were immediately improved and quickly forced Hull back into their own half.
For a while, they defended desperately but Spurs’ new-found confidence invited a quick counter.
Substitute Robert Koren nearly took advantage when his through-ball found Jelavic and, though he managed to stab his shot past the charging Lloris, Dawson was on hand to prevent it reaching the line.
Jelavic, who did everything except score on the day, struck the base of the post after more good link-up play from Long but the feeling remained that Hull were happier with the scoreline.
Had Adebayor had one of his better days, or Eriksen’s radar been better adjusted, Spurs would surely have scored again.
Hull might also have nicked it when Huddlestone’s flat free-kick in the 89th minute somehow evaded three team-mates inside the six-yard box, but neither side will be aggrieved with a draw.