Michael Bradley's side powered to a semi-final meeting with the Sale Sharks - conquerors of Agen this weekend - as the gained a 45-24 aggregate win over Grenoble - their sixth two-legged win over a French side in the last two years.
The westerners overcame the loss of veteran number 10 Eric Elwood to an eighth minute 'dead leg', to fend off Dean Richards' side - noted fast starters if last weekend's 18-3 lead after 26 minutes is anything to go by.
A smart Chris Keane break after seven minutes set up Australian Paul Warwick, who shifted from outside-centre to number 10 after Elwood's departure, for an 18 metre drop-goal.
After Tongan lock Alifeleti Fakaongo had knocked on with the Connacht try-line under threat, Lacey started and finished a fine 21st-minute forward drive, instigated by Warwick's huge cut out pass.
That score went unconverted and so too did Swift's effort after 32 minutes, following Keane's searing midfield break.
France international David Aucagne then snapped the deficit back to 13-3 at the break with a late 35-metre penalty.
If Warwick, Connacht's undoubted star of the tournament with 82 points in four starts, was off-colour with his right boot in the first half, the 24-year-old eased their nerves in the second as two penalties after 54 and 77 minutes - completing his nine-point haul - eased the Irish province through.
By that stage Grenoble fly-half replacement Maxime Suarez had received a straight red card from Welsh referee Hugh Watkins for a reckless head butt, but it mattered little to the off-the-pace French.
Afterwards, Connacht coach Bradley commended a 'job well done.' "We had a job to do and we are happy with the overall result and the way we played," he said.
"We got a lot of control on the game, more so than they did, and that was a key element for us to win the the game."
"It would have been very disappointing to have come off the park having not won the match. The only way that we would have lost is if we did not play well. That did not happen and we are very delighted with the win," he added.
"It is very important in terms of looking forward and going forward, and also from a profile point of view that we are competing in the latter stages of the competition."
Grenoble boss Richards said: ""We feel that we under-performed at this stage. We would liked to have given Connacht a better run for their money, but you cannot take it away from them either."
* The Challenge Cup semi finals will take place in April - with the first leg on April 1/2/3, and the second one on April 22/23/24. French sides Pau and Brive meet in the other semi final.
Saturday, January 15:European Challenge Cup: Quarter final, second leg:Connacht 19 Grenoble 3, The SportsgroundScorers: Connacht: Tries: Matt Lacey, Michael Swift; Pens: Paul Warwick 2; Drop goal: WarwickGrenoble: Pen: David Aucagne
(15) Matt Mostyn(14) Darren Yapp(13) Paul Warwick(12) James Downey(11) Ted Robinson(10) Eric Elwood(9) Chris Keane(1) Ray Hogan(2) Bernard Jackman(3) Peter Bracken(4) Christian Short(5) Andrew Farley (Capt)(6) Michael Swift(7) Matt Lacey(8) John Muldoon
Replacements used: Conor McPhillips for Elwood (8 mins), Adrian Clarke for Bracken (50), John O''Sullivan for Swift (59), Tom Tierney for Keane (68), John Fogarty for Jackman (71), David Slemen for Mostyn (78), Pieter Myburgh for Farley (80).
Red card: Maxime Suarez (Grenoble) (69 mins)HT: Connacht 13-3; Attendance: 2,500Referee: Hugh Watkins (Wales)