Blackpool's Gilks pulled off a flying save to keep out Hull midfielder David Meyler's goalbound shot in the 60th minute, while opposite number Jakupovic was at full stretch himself to turn away a vicious 25-yard effort from the home side's coveted Thomas Ince in the 85th minute.
Jakupovic, a Bosnian-born Switzerland international, then made another fine save in the second minute of time added on to keep out Seasiders' substitute Nathan Delfouneso's shot and ensure Hull stretched their unbeaten league run to eight matches.
Hull, who had won six of their previous seven npower Championship matches, remain second in the table, while Blackpool, for all their late pressure, have now won only one of their last six league matches.
Midfield pair Ludovic Sylvestre and Isaiah Osbourne, plus defender Neal Eardley returned to Blackpool's starting line-up, while Liam Rosenior stepped in at right-back for Hull in place of the injured Ahmed Elmohamady.
Neither side could fashion a chance worthy of note in the first 20 minutes save for Hull's on-loan Sunderland midfielder Meyler, who dragged an angled shot from the left side of the penalty area off target.
The visitors threatened again in the 22nd minute when striker Sone Aluko flashed a 25-yard shot wide and it was not until the 29th minute that Blackpool's coveted winger Ince fired his side's first shot on goal straight at Jakupovic.
The Seasiders enjoyed their best spell from the half-hour mark, with striker Gary Taylor-Fletcher and Ince both prominent, but failed to test Jakupovic again before the half-time whistle.
Hull manager Steve Bruce replaced Aluko and midfielder Robert Koren for strikers Nick Proschwitz and Jay Simpson at the interval and the latter was straight into the action when his header from Rosenior's cross forced Gilks into a smart save 90 seconds after the restart.
Gilks then held Hull midfielder Robbie Brady's 20-yard free-kick as the visitors went in search of the breakthrough.
Meyler, who has scored three goals in his last four appearances for the Tigers, went closest to the game's opening goal with a superb strike in the 60th minute that prompted a spectacular save from Gilks, whose outstretched arm turned the goalbound shot away for a corner.
Blackpool were dealt a blow soon after when striker Wes Thomas was forced off injured and replaced by Delfouneso before Sylvestre had a rare shot on goal for the home side, firing an angled drive just wide.
The game was finely balanced heading into the final quarter when Blackpool defender Chris Basham fired a 25-yard effort over the crossbar, while at the other end Hull midfielder Corry Evans spurned a gilt-edged chance when he failed to keep his shot from inside the area on target.
Jakupovic flung himself at Ince's 25-yard shot in the closing stages to push the ball away to safety before Delfouneso blazed over the crossbar from inside the area in stoppage time.