"I am really delighted, we are almost guaranteed a home quarter-final," said Harlequins director of rugby Conor O'Shea.
"We will enjoy tonight and have the opportunity of looking forward to that quarter-final at the Stoop on a hard pitch in April.
"Connacht came to play rugby; it was tough for long periods especially as we played into a stiff breeze in the first half.
"The penalty try just before half-time gave us breathing space. We were far more direct in the second half and played some superb rugby."
Ugo Monye celebrated his 200th game for the Quins with a splendid solo effort to collect the bonus point which drew a special mention from O'Shea.
"Ugo's try gave me the most pleasure of all," he added. "He has had a heck of a lot of good offers from other clubs but he has remained with us and he still has a lot to offer. I'm delighted for him."
Connacht head coach Eric Elwood said: "We had some great opportunities in the first half-hour and didn't have any reward. We had four or five great line-breaks and didn't finish them off.
"How we were 6-3 down after 35 minutes I just don't know but Quins scored just before and after half-time to build up a cushy lead."