The leading goalscorer in the third tier tucked home from close range after half an hour, before Matty Pearson's cross beat everyone including goalkeeper Steve Arnold as he found the back of the net shortly after the restart.
Collins rounded off the scoring for the afternoon when he managed to get an effort away from close range with 69 minutes on the clock to ensure the Hatters' unbeaten run continued.
The Town were unchanged for the fourth game in a row as they looked to make it 18 matches without defeat.
That meant James Shea was in goal behind a back four of Jack Stacey, Pearson, Sonny Bradley and James Justin.
The midfield was made of Alan McCormack, Pelly-Ruddock Mpanzu, Andrew Shinnie and Luke Berry, with a front two of Kazenga LuaLua and Collins - who was returning to the club with whom he won two promotions from League Two, and to a stadium where he remains the record goalscorer.
New loan signings Jason Cummings and Alex Baptiste took their place on the substitutes' bench, alongside Danny Hylton on his return from a four-match suspension.
Shrewsbury made the brighter start and Shea had to field a Greg Docherty shot from distance, before Bradley and Pearson dealt well with deep crosses into the box from both flanks.
The Hatters grew into the game and started to dominate possession, without finding the killer pass into the box, but it was Docherty who had the next attempt on goal in the 21st minute, firing over the bar after Shea could only palm Fejiri Okenabirhie's shot back into the danger area and Stacey's half-clearance went straight to the midfielder 12 yards from goal.
Stoke loanee Tyrese Campbell, son of former Arsenal striker Kevin, went close with a cheeky backheel soon after, as Shrewsbury continued to make all the running.
The breakthrough came with the Hatters' first attempt on goal on the half-hour mark though, and inevitably it was Collins - with his 17th goal of the season - who was the executor.
Bradley, Justin and Mpanzu traded passes on the Town left, around halfway, waiting patiently for the opportunity for the full-back to set off on one of his trademark raids.
The ball was worked to McCormack, who crossed to the far post, where Berry had wandered into the box to lay a header back across goal for Collins to steer into the back of the net to the delight of the travelling fans behind that goal.
The Town started to control proceedings, and almost doubled the advantage on the stroke of half-time when LuaLua crossed from the left and almost picked Justin out as he raced towards the near post, but Shrews keeper Arnold stood up to keep the cross out.
The hosts started the second half as brightly as they did the first, but couldn't capitalise before Pearson netted the Hatters' second, somewhat fortuitously, but brilliantly nonetheless.
Justin's deep free-kick from the left beat everyone, but as Shrewsbury struggled to clear from the left-back position, Stacey fed Pearson on the touchline, and the centre-half's cross drifted over Arnold's head and inside the far post for his fifth goal of the season.
Shinnie might have sealed it in the 66th minute as LuaLua slipped him in to the left of the area, but the Scot's shot was straight at Arnold.
Three minutes later the killer strike did come, however, and it was Collins' second, digging his shot out from the edge of the six-yard box at the third attempt after LuaLua had teed him up, that did the damage.
Hylton came on for LuaLua, who had run himself into the ground once again, and almost scored with his first touch, Arnold saving well as the striker directed a header from Stacey's right-wing cross towards the bottom corner.
Justin soon forced Arnold into another flying save with a curling free-kick, with Shrewsbury limited to an Okenabirhie free-kick that flew high into the 1,334 Town fans behind the goal, before Luke Waterfall sent a close range effort the same way.
It had been a tougher afternoon than the final scoreline would suggest, but the Hatters' snowball continues to roll with the club record 19-game unbeaten run in the Football League now just one result away.
TOWN: Shea, Stacey, Pearson, Bradley (c), Justin, McCormack (Potts 78), Mpanzu, Shinnie, Berry (Moncur 83), Collins, LuaLua (Hylton 70). Subs: Baptiste, Cummings, Sheehan, Isted
Goals: Collins 30, 69, Pearson 53
Yellows: Mpanzu
SHREWSBURY: Arnold, Sadler (c), Docherty, Campbell (Payne 63), Okenabirhie, Bolton, Vincelot (Whalley 55), Waterfall, Golbourne, Norburn (Laurent 76), Grant. Subs: Mitchell (GK), Williams, Holloway, Sears.
Yellows: Grant, Docherty, Golbourne
REFEREE: Graham Salisbury
ATT: 6,859 (1,334 Hatters)
