A 9-8 defeat at a wet and windy Sportsground has seen Quins finish second in the pool, just a point behind four-time champions Toulouse - who suffered a surprise defeat at Gloucester.
Having taken a 9-5 lead into the break, thanks to three penalties from Niall O'Connor, Connacht defended for their lives in the second half and frustrated Quins into submission.
Sam Smith's early try was the high point for Conor O'Shea's men, who could only add a single penalty from Nick Evans after that.
The windy conditions were a factor when Evans missed a late kickable chance and the ball also squirted loose from a second half scrum as the Quins forwards battered away at a stubborn Connacht defence.
Speaking afterwards, Quins captain Robshaw said: "Playing into the wind, you always have to play a bit more with the ball and to only go down by four points at half-time, it was kind of ideal so to speak.
"We came out buzzing, got our feet on the ground, got a bit of momentum going but unfortunately we couldn't finish it off. It's hard to put into words how disappointed we are, it's a very low point."
The Aviva Premiership leaders could still sneak into the last-eight of the Heineken Cup, but they would require an unlikely run of results in the other pools over the weekend.
Asked if the difficult conditions were a decisive factor, Robshaw insisted: "The conditions were the same for both sides, they probably played them a little better than us.
"We always knew it was going to be hard going into the wind. They're probably a bit more used to it than us, but that's no excuse.
"We're going to have to console ourselves tonight, look back on the video and hope that someone else slips up along the line. It probably doesn't look like (we'll qualify).
"No-one was complacent. We wanted to be in the knockout stages. We didn't just want to be there to make up the numbers - we came in here to give it a real go.
"Obviously we'll see where things end up on Sunday night, but in all honesty we probably will be back in the Amlin (defending the title we won last year)."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.