Niall O'Connor gave the hosts the lead with a penalty kick after three minutes.
The visitors were soon ahead as Sam Smith touched down on seven minutes. However, Nick Evans was wayward with his kick in windy conditions.
O'Connor put Connacht back in front with a penalty on 15 minutes before another kick a minute later gave the Irish side a 9-5 lead at the break.
After a tense opening to the second half, Evans got Quins back into the tie with a penalty kick.
As time wore on, a score would have taken the English side through to the quarter-finals. However, Connacht stubbornly held on to claim their first ever win in the tournament.