Elliott joined United on Sunday following a hugely successful spell with Barnsley-based Shaw Lane.
Mr Newton told bufc.co.uk: "We are delighted that Craig Elliott has joined Boston United. His track record as a manager is second-to-none within the non-league system over the past few years, which has seen him take Shaw Lane to the top of the Evo-Stik Northern Premier Division with three promotions in four seasons.
"Not only has he been successful, but has been consistently successful.
"Craig is a very ambitious, young manager who has a clear vision of where he wants to go within his career and his ambitions match ours.
"None of us are in doubt about the challenges ahead, but we will support Craig as far as we possibly can to ensure that things progress on the pitch as quickly as possible.
"My sincere thanks go to Karl Hawley who stepped in to take charge of the first team following Adam Murray's departure. The team have put in three fantastic performances under Karl's guidance and we will always be very grateful to Karl for what he has done for the club to date.
"My thanks must also go to Craig Wood, the chairman of Shaw Lane. Understandably Mr Wood did not want such a successful manager to leave his club, but also did not wish to stand in the way of Craig's career progression.
"I was delighted that we were able to reach an agreement that was mutually acceptable to both clubs and I wish them all the very best with their promotion push.
"Craig inherits a really committed squad and I hope they continue to progress and pick up not only the much-needed points that the club requires, but also the points that their commitment deserves."
Updated 21:00 - 12 Nov 2017 by Craig Singleton