The Spanish striker showed stunning footwork to slot in a first, before Geovanni's volley levelled matters.
Geovanni headed straight at Jaaskelainen from 10 yards and those misses almost came back to haunt Hull.
Dean Marney sidefooted a shot across goal after Jaaskelainen spilt a Geovanni shot as Hull continued to attack.
The desperation of their predicament was mirrored in their tenacity and Geovanni could have given them an early lead.
Brazilian Geovanni made it 2-0 with a diving header after Marlon King's flick before King slotted in a low shot.
But Hull hit back through Geovanni's brilliant 25-yard strike on 61 minutes before Cousin's well-placed glancing header from Andy Dawson's corner.
But the home side kept fighting and were rewarded with Geovanni's goal from Daniel Cousin's cross to set up a frantic finish.
Geovanni slotted in to revive Hull's hopes of a draw but Kuyt hit his second when Alvaro Arbeloa's shot fell to him.
Geovanni opened the scoring for the Tigers, but his ninth-minute effort was cancelled out by Beijing's Mobido William just after the break.
First, King produced a superb volleyed flick for Geovanni to run onto and meet with a diving header which left Carson with no chance.
Soon after, Marney latched on to a fine throughball from Geovanni but dragged his shot across the face of the goal and inches wide.
Hull might have had their third when Geovanni had three attempts at a free-kick, with two Manchester City players booked for encroaching from the wall.
But Geovanni equalised against his former club with a deflected free-kick.
Geovanni swept home the Tigers' second goal from Kamil Zayatte's cross but Scott Sinclair's late header for Wigan left Hull hanging on at the end.
And his anxiety almost increased after four minutes when Geovanni slipped in behind Woodgate, but his finish was uncharacteristically wild as he fired over the top.
Eight minutes later, the game was beyond Wigan when Geovanni's volley rounded off a move that started when Emmerson Boyce gave the ball away to Zayatte.
Hull responded by pouring forward again and went close to breaking the deadlock when Geovanni curled in a cross that Vennegoor of Hesselink stooped to head inches wide.
Hull responded with a Geovanni piledriver that flew past the angle of Jose Reina's goal and a Liverpool defence that has conceded eight goals in their last two matches was looking shaky.
Hull boss Phil Brown made two changes from the side that drew 2-2 with Everton last week, with recalls for midfielders George Boateng and Geovanni, while Peter Halmosi and Bernard Mendy dropped to the bench.
Meanwhile, City's new signing, Brazilian international Geovanni, has joined his new team mates at the club's Italian training camp in the foothills of the Alps after agreeing a two-year contract with the Premier League newcomers.
Geovanni showed true Brazilian flair with an aerial volley that flew narrowly over the crossbar but he pulled Hull level on the hour mark when his free-kick found the net after a deflection off Vincent Kompany.
One of their better openings in the first half came when a deep-lying Geovanni floated a bouncing ball over the top for Martin Petrov to accelerate onto, only for on-rushing Wigan keeper Kirkland to clear decisively.
With the game stretched and there to be won by either team, City boss Sven-Goran Eriksson made a defensive-looking substitution, withdrawing the fading Geovanni in his free role for Kelvin Etuhu to make a five-man midfield.
But Hull regained parity with a sensational strike from Geovanni, who skipped past a challenge on the left before drifting infield and smashing a right-foot shot that thundered into the top left-hand corner of Manuel Almunia's net.
At that stage the score was harsh on Hull, who saw a Sonko header deflected wide and a Geovanni shot fly off target, but from then on it was nothing more than an exercise in damage limitation.
Quaresma had a spell with Barcelona earlier in his career, but Hull also had a former Barca player in their ranks in Geovanni and the Brazilian fired a sublime half-volley inches wide of Henrique Hilario's goal, with Chelsea's stand-in keeper stranded.
Paul McShane's cross from the right was headed back across goal by Jan Venegoor of Hesselink and, after Martin Skrtel could only flick the ball up in the air, Geovanni waited for it to drop on to his left foot and directed it into the corner from 10 yards.
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