Jones made his decision to take over at the bet365 stadium just days before the Hatters head to fellow League One promotion chasers Sunderland and then have a FA Cup third round replay against Sheffield Wednesday, with the winners heading to Premier League giants Chelsea.
An upbeat Sweet insisted though that the club could be even stronger due to the departure, saying: "I can understand why they might feel down temporarily, but there’s no reason to feel down.
“One thing I always say is it doesn’t matter who it is in the business, if you have a departure, it’s an opportunity to improve.
“Somebody leaves, there’s always an opportunity to get better, that’s our objective.
“So don’t feel down, I understood why people felt emotionally attached to Nathan, because he wore his heart on his sleeve, he’s not wearing that heart now, lets move on, lets focus on the future.
"What isn't there to be excited about? This is a hugely exciting time for Luton Town football club, don’t let one person affect your loyalty or your judgement of that, we're all together in this.”
Sweet moved swiftly to replace Jones for the short term, with club legend and ex-player Mick Harford installed as interim manager.
The Town CEO believes that with the former Town boss in charge, he can help Luton on and off the pitch in their hunt for new supremo.
He continued: “We’ve got Mick at the helm, who wouldn’t want that?
"With Mick and a couple of others guiding us with regards to finding a new recruit, we’re fully confident that we’ll be able to find someone who is more than appropriate and suitable for the Luton promotion campaign for this season.
"We really genuinely don’t think that will be derailed, so get yourselves up and gets yourselves up to Sunderland.
"Lets support this man here and lets support the 11 on the pitch, lets get three points and lets get Sheffield Wednesday on Tuesday night under the lights at Kenilworth Road, with Chelsea down the line."
A number of Town fans have were disappointed that Jones opted to leave the club after continually expressing his love for the job he was doing at Luton, plus his passionate post match celebrations.
However, Sweet knows full well that football can be a fickle game, saying: "I can also understand why supporters felt aggrieved about it, and principally because he’s a passionate manager who wears his heart on his sleeve.
"He was quite animated with supporters and truly showed that emotional side and what he called the love of Luton Town football club, and supporters believed it.
"I'm not saying that he didn’t mean it at the time, I’m sure he did, but supporters believed that and there is very, very little loyalty in football, unfortunately, as we know.
“You will get it from people like myself and our board, because we are Luton, we are madly in love with our football club and our town and we will never go, but you don’t get managers like that.
“I’ve got a man sitting next to me (Mick Harford) who's as close as you’ll get.
"He might be a Mackem, but he’s a Luton man, and so this is as close as we can get to a manager who’s going to manage our team for the temporary period, who is going to be as loyal and faithful to the football club as us.”
Harford himself anticipates the club will pull together in the wake of Jones' exit, as he when asked if it was like breaking up with a loved one, he said: "It’s not as bad as that, I’ve lost a few girlfriends!
“Obviously we’ll get over it and we’ll face the challenge of appointing a new manager.
"I’ve got to be honest with you, I’m totally privileged that Gary and the board and asked me to be involved in the process of trying to find a new manager, I’m privileged to be in that position."