Nathan Jones made two changes to the side that lost 1-0 at home to Preston three days previously, with Cauley Woodrow making his first Luton start in the league and Gabe Osho coming into midfield, Jordan Clark and Luke Freeman being named among the substitutes.
Admiral Muskwe was also on the bench, having missed out on Saturday, with Louie Watson the man to drop out of the matchday 18.
City had the lead inside five minutes, when Wells timed his run perfectly to latch onto a through ball from Alex Scott and finish across Ethan Horvath.
Elijah Adebayo had the Town's first sight of goal on 16 minutes, collecting a short pass from Allan Campbell 30 yards out and taking a couple of strides before arrowing a shot that went just wide of Dan Bentley's right post.
The hosts doubled their advantage in the 27th minute when Andi Weimann raced onto a Han-Noah Massengo pass down the left and crossed low for Conway to slam the ball in from close range.
Dan Potts blocked another effort from Conway just after the half-hour, then Massengo saw a 25-yarder deflected wide for a corner as City moved the ball about and looked to increase their lead.
Gabe Osho won a crunching challenge in midfield to set Adebayo free on the right, and his low cross intended for Carlton Morris was turned behind, with last season's top scorer climbing high to mee James Bree's corner, but unable to keep his header down.
Jones resisted the option to make a double change before the break, despite Clark and Freeman being readied, and instead waited until the interval to bring on three; the midfield duo on in place of Potts and Woodrow, with Admiral Muskwe coming on for Reece Burke.
That meant a switch to a 4-3-3, with Osho dropping to centre-half alongside Bradley and Bree and Amari'i Bell playing as full-backs.
City were still the side creating chances, however, Conway breaking through the backline and firing straight at Horvath, before Weimann skied a 54th-minute shot from just inside the area.
Jay Dasilva was next to cause a moment of danger, skipping his way inside the Town penalty area from wide on the left and seeing a right-footed shot deflected behind off Bradley.
Rob Atkinson went down in the box under challenge from Clark, but referee Lee Doughty waved appeals for a penalty away, only to soon be at the centre of a call that reduced the hosts to ten men.
Mark Sykes lunged in on Freeman, for which he would eventually be shown a red card, and in the melee that followed, went down holding his head when the Town midfielder clearly barged into his back with his chest. Freeman and Bradley were both shown yellow, along with Weimann.
Town looked to make the extra man count, and Campbell burst through into the City penalty area, only to see his cross flash straight across the six-yard box, then Clark shot high over the bar from Adebayo's cut back.
Cameron Jerome replaced Morris with just over ten minutes left, then Bree sent a free-kick inches over the bar before Elliot Thorpe was introduced in place of Campbell.
Osho twice could have reduced the deficit in added time but for a brilliant low stop from Bentley on the second occasion, as there continued to be no way through the Robins' rearguard.
City therefore recorded their first victory of the campaign and left the Hatters still seeking theirs after four matches.
GOALS:
BC - Wells 5, Conway 27
