The striker followed up his goal in our Under-21s fine win against Chelsea on Friday with a treble of lethal finishes, all dispatched in one-on-one situations.
Darren McQueen headed home our second goal on the night as the team - including six starters from that U21 victory - put on a fine display against the Conference South side.
Jonathan Obika also returned from an ankle injury as we came face-to-face with two former Spurs trainees in the Chelmsford line-up, goalkeeper Tommy Forecast and midfielder Mark Hughes.
Forecast made sure it wasn't a landslide with a string of outstanding saves in the first half but the home team had their moments as well, fighting back to 1-1 in the 17th minute and then forcing Jonathan Miles into a fine save to maintain our 2-1 lead at half-time.
Forecast's first save came in the fifth minute to turn aside Cameron Lancaster's curler and Lancaster soon turned provider for the opening goal.
Picking up the ball centrally, he slid an inch-perfect pass into the path of Coulthirst bursting into the box and the striker made no mistake from the right angle, 1-0 on 13 minutes.
Chelmsford hit back four minutes later when Dominic Ball was penalised for handball and Jeff Goulding make no mistake from the spot.
It was then all about Forecast, who took off at full length to deny Coulthirst and Lancaster in quick succession and when he was beaten, Darren McQueen's low shot hit the inside of the post and cannoned to safety.
McQueen didn't have to wait long to register though and made it 2-1 when he rose unchallenged at the far post to head home Aaron McEneff's free-kick on 27 minutes.
Miles then made his save from Bagasan Graham and the last word went to Forecast, who stood firm to keep out Obika's close-range effort.
We made four changes at half-time with Lawrence Vigouroux, Alex McQueen, Ruben Lameiras and Roman Michael-Percil on for Miles, Ryan Fredericks, Lancaster and Darren McQueen.
Goulding fired over an early chance for City before Forecast blocked Obika at the near post.
The match was relatively quiet then until the midway point of the half. Laste Dombaxe came off the bench to replace Giancarlo Gallifuoco and almost scored with his first touch, a 30-yard free-kick that tested Forecast again.
It was 3-1 on 71 minutes when Coulthirst raced onto McEneff's pass through the middle and finished with some style.
He was there again two minutes later to complete his hat-trick and make it 4-1, a low finish past Forecast from the left side of goal.
Vigouroux went full stretch to tip over Bertie Brayley's downward header on 76 minutes and the final action of note saw Obika denied again, this time Matt Lock back just in time to clear his goalbound effort off the line.
Chelmsford City squad: Forecast, Miller, Redwood, Davis, Haines, Webb. Paxman, Hughes, Goulding, Derry, Graham, Long, St Aimee, Nicolau, Rainford, Brayley, Holdsworth, Pentney, Spriggs, Sanparo.
Spurs XI: Miles (Vigouroux, 46); Fredericks (A McQueen, 46), Ball, Stewart, Ward; Gallifuoco (Dombaxe, 68), McEneff, Lancaster (Lameiras, 46); D McQueen (Michael-Percil, 46), Obika, Coulthirst.