Howard slid the ball home in the 72nd minute to seal the win after Town had struggled to hit the target despite creating a string of chances.
For Rams' caretaker boss Westley, previously a youth coach and manager at the Hatters in the 1990s, it meant a continuation to County's Championship relegation battle.
Meanwhile McIndoe, a former Town trainee, suffered a miserable afternoon and was substituted on the hour.
A dour opening saw Sol Davis have a shot blocked by Darren Moore in the fifth minute, but Carlos Edwards' shot that flew well wide from Ahmet Brkovic's mis-hit cross 15 minutes later summed up the quality on offer early on.
Town were the first to get a grip on proceedings and, in the 22nd minute, Warren Feeney played a one-two with Howard, wriggled clear of Moore and crossed for Edwards to take a touch past Lewin Nyatanga before firing just past the post left-footed from 12 yards.
Brkovic then fired just past the opposite post from 25 yards on the half-hour mark after Davis had beautifully controlled a Russell Perrett headed clearance and found the Croat on the edge of the penalty area with a measured pass.
Derby had showed little, but gave an indication of their threat on the break in the 34th minute when Giles Barnes set off on a 40-yard burst before testing Marlon Beresford with a low 25-yard drive.
Rams goalkeeper Lee Camp then produced his first save of the game in the 36th minute as he blocked at the feet of Feeney eight yards out after Howard had just failed to connect with an Edwards centre from the right.
Kevin Foley should have at least worked Camp again four minutes before the break when he volleyed wide right-footed from 15 yards after Feeney had nodded back a Paul Underwood cross from the left.
Brkovic then volleyed from almost the exact same spot three minutes later after another Edwards cross, but again the ball veered off target as it flew over.
Luton's first-half dominance continued as Edwards broke clear after a Derby corner and twisted Alan Wright inside out before unleashing a 20-yard shot that was parried by Camp.
The visitors improved after the interval as Moore sent two headers just off target firstly from an Inigo Idiakez free-kick in the 49th minute and then a corner from the same player two minutes later.
Derby's Charlton loan man Kevin Lisbie produced his first effort of the day just before the hour mark as he received Wright's throw-in, turned Perrett and shot wide from 18 yards.
Luton regrouped and wasted two glorious chances within the space of three minutes. Firstly Feeney beat Wright to Beresford's 66th minute clearance, raced into the box and unselfishly pulled back for Brkovic to blast over from seven yards.
And moments later Feeney was this time the culprit as he controlled Underwood's cross, turned Wright, but smashed his eight-yard shot across the face of goal and past the far post.
Howard showed everyone how it should be done though in the 72nd minute as yet another Edwards cross was flicked on by Coyne for the burly striker to slide the ball home at the far post and finally break the deadlock.