Highly-rated forward Powell illuminated the League Two play-off final to give Crewe the lead on 15 minutes.
The England Under-18 international - one of nine products from the famous Gresty Road academy in Steve Davis' starting line-up - is reportedly the subject of transfer interest from host of Premier League clubs, and his spectacular turn and shot after a quarter of an hour will have done his prospects of a big-money move little harm.
Alex made sure of promotion after three seasons in the fourth tier when Moore opened up a flagging Robins backline eight minutes from time.
But Crewe were made to work hard in blistering temperatures before chalking up a 19th game without defeat.
Jeff Goulding almost had Cheltenham back on terms when he rattled the crossbar moments after Powell's opener, before Alex somehow survived an onslaught of set-piece pressure on the stroke of the interval.
Mark Yates' men failed to maintain the charge in the closing stages as the succumbed to a first play-off defeat in their history.
Cheltenham made two changes to the side that overcame Torquay in the semi-final second leg 10 days ago as Keith Lowe and James Spencer dropped to the bench to accommodate Everton loanee Luke Garbutt and Goulding.
Luke Murphy recovered from a groin injury to take Lee Bell's place in the Crewe midfield.
Following a sedate opening to the game, Powell put the Railwaymen ahead in sublime fashion.
Collecting a pass from Kelvin Mellor, he produced a clever flick to escape the attentions of Marlon Pack and Alan Bennett before rifling an unstoppable left-footed shot beyond Brown from 20 yards.
Cheltenham almost found an immediate response as Goulding crashed a venomous long-range effort against the bar, Ben Burgess unable to convert the rebound as goalkeeper Steve Phillips remained rooted to his line.
Goulding converted a Burgess flick-on from an offside position in the 25th minute and the latter's aerial prowess was starting to cause problems for Crewe.
Kaid Mohamed and Goulding both shot wastefully over following knockdowns from the former Blackpool man, and Powell almost made them rue their profligacy when he scampered in from from the left to fire narrowly past the far post in in the 36th minute.
Mellor rose unchallenged to plant Ashley Westwood's 41st-minute corner over the bar, but Cheltenham's set-piece threat brought them desperately close to an equaliser before the break.
Amid a flurry of intense pressure, Phillips was at full stretch to deny Goulding before Powell and Westwood cleared off the line from Steve Elliott and Mohamed respectively.
Cheltenham retained that intensity as the second-half got under way, with Pack fizzing a low 20-yard drive just wide.
An untimely stumble from Crewe centre-back Adam Dugdale allowed Mohamed to scamper beyond him and through on goal in the 54th minute, but Phillips stood firm to save.
Dugdale found himself in the thick of the action in Town's area with 19 minutes to play, but a heavy touch at the back post from Westwood's free-kick allowed Brown the time to produce a fine last-ditch stop.
Moore saw a shot from the edge of the area squirm wide before making amends by clinching League One football in the 82nd minute.
The winger exchanged passed with Ajay Leitch-Smith and steered low into the far corner despite losing his footing.
Brown spared his side further woe with a pair of splendid late saves from Moore and substitute Max Clayton.