The former Republic of Ireland under-21 international has penned a three-year deal and moves to Easter Road as a direct replacement for last year's double player-of-the-year-award winner Leigh Griffiths.
The successful bid - reported to be worth around B£200,000 - will come as a major boost to the Hibs support after the Leith side suffered a humiliating 9-0 aggregate defeat to Malmo in the Europa League second qualifying round.
The 22-year-old said: "I'm really excited about joining Hibs and playing in the Scottish Premiership.
"It will be a new challenge for me and I'm aiming to adapt quickly and to score goals for the team.
"Once the bid to Swindon was accepted, the manager showed me the stadium and training centre and it was a real eye-opener; I didn't realise just how big the club is.
"So it was a really easy decision for to make and now I'm just targeting a good run of games in the side and to score some goals."
Originally with Aston Villa, Collins established his reputation as a proven goalscorer with spells at Shrewsbury and Swindon and says he feels equipped for the task of making an impact in Scotland's top flight.
He added: "I'm moving from the third tier in English football to the top tier in Scotland, which is quite a big step, but I'm confident I can make an impact and settle in quickly."
Capped at both under-19 and under-21 level by the Republic of Ireland, Collins says he is hoping scoring goals regularly for Hibernian could ultimately earn him a call-up into the senior squad.
He said: "That would be a dream come true if I was called up, but first and foremost I need to start well with Hibs, stay fit and score goals.
"I'm going to give everything I can to help the team to do well in the Scottish Premiership and I'm looking forward to playing in front of the Easter Road crowd."