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.
Quaresma had a lively start to the game and saw a fine curling shot tipped around the post by Hull keeper Matt Duke.
Soon after, Ballack hit the outside of the post with a 25-yard free-kick as Chelsea continued to press.
But the visitors always looked dangerous on the break and Kevin Kilbane was unlucky to see his header from 12-yards hit the post.
Commendably, after the break, rather than sit back and plant men behind the ball, Hull went in search of the winner.
Brown happy to settle for draw
Fagan found himself one-on-one with Hilario but tamely chipped the ball into the hands of the Chelsea keeper.
Soon after, Marney latched on to a fine throughball from Geovanni but dragged his shot across the face of the goal and inches wide.
Didier Drogba was introduced but it was Salomon Kalou who had Chelsea's best chance of the second half.
The Blues forward found himself with just Duke to beat but from 18-yards out he shot tamely at the Hull keeper.
Referee Lee Mason also waved away penalty appeals from Chelsea when Andy Dawson appeared to handle the ball as their frustrations grew.
Chelsea lucky to escape with draw - Wilkins
Ian Ashbee had a late chance for Hull from a corner, but he fired wide from close range.
The final whistle brought a chorus of boos around Stamford Bridge as Chelsea were left to reflect on the 16 points they have dropped at home this season.
One banner unfurled by fans called for Scolari's sacking.
Hull are still without a win in nine league games, but this was a hugely encouraging point for Phil Brown's side and their performance was more reminiscent of their early season form when they won away at Tottenham and Arsenal.
Chelsea: Hilario, Bosingwa, Alex, Terry, Ashley Cole, Mikel (Belletti 57), Quaresma (Drogba 63), Ballack (Deco 73), Lampard, Kalou, Anelka.
Subs Not Used: Taylor, Ivanovic, Di Santo, Stoch.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.