Bugiel scored from the penalty spot to complete his hat-trick and Worthing will now travel to division-lower Carshalton for the replay on Monday evening.
Worthing went into the game looking to bounce back from bank holiday weekend defeats to Grays Athletic (1-0) and the 5-0 hammering at local rivals Bognor Regis but trailed 2-1 at half-time.
Carshalton added a third on the hour but were then reduced to ten men when Sekou Kaba was sent off with 23 minutes left. Worthing then pressed forward and Bugiel netted twice - including his third from the penalty spot with virtually the last kick - to keep his side in this season’s competition.
The hosts never really got going in the first half, although they did go ahead on 21 minutes. Reis Stainslaus slipped Bugiel through and he beat Kalvin Morath-Gibbs before firing low past Carshalton keeper Alex Kozakis.
Worthing keeper Rikki Banks made a smart stop to keep out Kaba, before Worthing went agonisingly close to a second on 31 minutes. Sam Rents’ free kick picked out Stanislaus at the back post and his volley was pushed on to the bar from point-blank range by Kozakis.
The visitors equalised on 38 minutes when Mickel Miller raced on to Kaba’s pass, burst into the penalty area and fired low into the far corner.
Just three minutes later, Carshalton went ahead. Worthing’s defence stood still waiting for the offside flag and Kaba raced on to a ball over the top and then flicked it past Banks to Tommy Bradford, who headed home into the empty net.
Carshalton were the first to threaten in the second half when Miller drove a low shot straight at Banks from the edge of the penalty area.
Bugiel shot a daisycutter straight at Kozakis on 55 minutes from 20 yards, before Carshalton had a great chance to increase their lead three minutes later. Kaba slipped Ricky Korboa through on goal but he weakly side-footed straight at Banks from eight yards.
Carshalton did go 3-1 ahead on 61 minutes when Korboa’s low cross found Miller, whose shot was brilliantly saved by Banks but Miller then tucked home the rebound.
The hosts threatened to get a goal back on 66 minutes when Bugiel’s shot took a deflection and was stopped on the line by Kozakis.
Just a minute later, Carshalton were reduced to ten men when Kaba was sent off for two bookings. Worthing began to control proceedings as they dominated possession and Harvey Sparks headed a Zack Newton cross over on 74 minutes, before they got a goal back three minutes later.
Newton crossed from the right and Bugiel got in front of his man to head home from eight yards.
Worthing had a excellent chance to level after 82 minutes when Gary Elphick rose highest to meet Sam Rents’ corner but headed over from eight yards.
The hosts forced several corners in the closing stages but Carshalton stood firm until the sixth minute of injury-time when Worthing were awarded a spot-kick for what appeared to be handball and Bugiel sent Kozakis the wrong way.
WORTHING: Banks; Hendon, Elphick, Metcalf (Brodie 78), Rents; Newton, Wills, Axell (O’Neill 71), Sparks; Stanislaus, Bugiel. Subs: Fagan, Edwards, Piper, Watson.
Don’t miss out on all the latest breaking news where you live.
Here are four ways you can be sure you’ll be amongst the first to know what’s going on.
1 Make this website your homepage
2 Like our Facebook page at facebook.com/pages/Sport-Sussex
3 Follow us on Twitter @SportSussex
4 Register with us by clicking on ‘sign in’ (top right corner). You can then receive our daily newsletter AND add your point of view to stories that you read here.
And do share with your family and friends - so they don’t miss out!