Michael Bradley's side - shorn of both full back Matt Mostyn (hamstring) and flanker Matt Lacey (thigh) shortly before kick-off - powered through to the province's fifth win on French soil as they held Dean Richards' Top 16 side scoreless in the second half.
Despite Paul Warwick, who went on to kick 16 points, lobbing over a third minute penalty, Grenoble were on top in the opening stages.
Jean-Victor Bertrand angled over for a counter-attack try only seconds later, as he struck off a David Aucagne and Yohann Authier set-up.
French international Aucagne missed the conversion, but added a penalty and the extras to full back Nicolas Carmona's 20th-minute breakaway effort, which was started by Rickus Lubbe's clever grubber kick.
Bradley's charges, on a three-game losing streak in the Celtic League, kept in the hunt at 15-3 down, as Warwick kicked a 32nd-minute penalty, sandwiched by two Aucagne efforts.
Crucially ex-Leinster wing Downey slalomed through the cover with two minutes remaining in the half to draw Connacht back to 21-13 down at the break.
Bradley threw Chris Keane and Christian Short on after the restart, as Connacht gradually assumed control. It paid dividends as they held Grenoble at bay in a magnificent display.
Warwick banged over his third penalty on 51 minutes, and just eight minutes later, the visitors hit the front as skipper Farley's barge-over try was converted.
Connacht's defence then held firm as they kept the Alps side out of their 22, and the home side's challenge wilted with second row Karim Ghezal yellow-carded in injury time.
Warwick clipped over a 15-metre penalty seconds later to confirm a five-point cushion for Saturday's second leg at the Sportsground (kick-off 2.00pm).
Afterwards, former England number eight Richards focused his thoughts on next weekend's vital clash.
"You can't win a match without keeping the ball. In the second half, we seemed to not know how to maintain the possession," said the 41-year-old former Lion.
"We'll go to Connacht to win and we have one week now to correct our errors. The priority remains the next match, and it's one we have to simply win."
European Challenge Cup: Quarter final, first leg:Grenoble 21 Connacht 26, Stade LesdiguieresScorers: Grenoble: Tries: Jean-Victor Bertrand, Nicolas Carmona; Con: David Aucagne; Pens: Aucagne 2Connacht: Tries: James Downey, Andrew Farley; Cons: Paul Warwick 2; Pens: Warwick 4
Replacements used: Chris Keane for Tierney (47 mins), Christian Short for Myburgh (52), Adrian Clarke for Ray Hogan (76), Niall O'Brien for Muldoon (88). Not used: John Fogarty.
Sin-bin: Karim Ghezal (Grenoble) (82 mins)HT: Grenoble 21 Connacht 13; Attendance: 3,200Referee: Roy Maybank (England)