Worcester needed a win to keep its hopes of topping Pool 5 alive and looked to be in control as it ran in four first-half tries to go in at the break 24-14 in front, with John Muldoon and Niyi Adeolokun crossing for the Irish province, which ended the half with 13 men.
However, it could only convert two of those touchdowns and that profligacy from the tee would come back to haunt the Premiership struggler, as Tiernan O'Halloran went over and Craig Ronaldson kicked five points to clinch the draw.
The result moves Connacht onto an unassailable 21 points at the summit ahead of the final round of fixtures, with Worcester's bonus point only enough to close the gap to six, although it could yet qualify as one of three second-placed teams.
Brive later pulled within three points of Worcester with a 29-19 win against eliminated Oyonnax.
Pau and Gloucester will face off for Pool 3's automatic qualification spot after the Top 14 side moved back to the top of the standings with a 42-14 demolition of Zebre.
The French club leads the Cherry and Whites by three points ahead of their clash at Kingsholm on Saturday (AEDT).
Sale Sharks' second win of the campaign - 15-13 at home to Lyon - opened the door for Cardiff Blues to potentially clinch Pool 2 with a win over Toulouse, while London Irish thumped Krasny Yar 47-17 in a clash between Pool 4's bottom two sides.
The day's remaining fixture saw second-placed Bordeaux Begles pull five points clear of the Dragons with a 36-28 win at Stade Chaban-Delmas, although Pool 1 looks set to be won by Newcastle Falcons when it hosts winless Enisei STM on Monday (AEDT).