An outstanding free-kick from Jesse Starkey and a debut goal from Mo Diallo had the Reds seemingly in control of the tie, but three second half goals from the hosts amid a third Worthing strike from Dajon Golding forced to tie into a penalty shootout, which the Wasps won by four spot kicks to three.
It was the home side who started the match the brighter, and a fumbled defensive header by Worthing debutant Sam Keefe gave Grinstead winger Jerry Amoo the first clear-cut chance of the game in the eighth minute, but his shot was comfortably saved by Jack Fagan.
Five minutes later, the hosts would have another chance to break the deadlock after a corner was met by Glenn Wilson, but his free header was off target.
Chances for either side were hard to come by until the 24th minute, when Ricky Aguiar was brought down on the edge of the area for a free kick to the visitors.
Aguiar, Joel Colbran and Jesse Starkey all stood over the ball, with the latter taking the eventual set piece after the first two dummied, which flew over a helpless Jacob Adams and into the top right hand corner to put Worthing 1-0 up.
The Reds then had a chance to double their lead four minutes later, when Mo Diallo surged inside the penalty area and punted an effort initially fumbled by Adams, but the goalkeeper gathered in the nick of time with Meekums closing in.
A mix-up at the back between Keefe and Aarran Raccine in the 32nd minute allowed Amoo to run clean through on goal, but Raccine recovered quickly to make a fantastic block to deny the hosts an equaliser.
Two minutes before half-time, Worthing doubled their advantage when Marvin Armstrong drove through the heart of the Grinstead midfield and fired a low shot which was parried by Adams, but Mo Diallo was on hand to smash the ball into an empty net to open his account on his debut.
Worthing had the chance to grab a third on the stroke of half time when Meekums ran to the byline and lashed the ball across the face of goal, but his cross narrowly evaded Jasper Pattenden at the back post meaning the score remained 2-0 at the interval.
The opening stages of the second period saw a big shout for a penalty when the ball struck the body of Colbran inside the area, but referee waved away the home side’s appeals.
In the 52nd minute, East Grinstead’s bright start to the half was rewarded when a cross from Jay Gassion was met by a diving header from Michael Uwezu initially saved by Fagan, but Louis Theophanous was on hand to tap in the rebound and half the Red’s lead.
Worthing instantly looked to restore their two-goal lead when Pattenden crossed the ball in from right, found Aguiar on the edge of the area who lashed his shot narrowly wide of Adams’ post.
Three minutes later, the hosts had the ball in the back of the net when Uwezu turned in another rebound after a fantastic save from Fagan, but the lineman’s flag was raised instantly to rule it out.
The home side’s misfortune showed no signs of stopping when Wilson picked up the ball and fired a dipping volley from 30 yards out which beat Fagan but cannoned back off the bar in the 63rd minute.
Worthing weathered the storm created by the Wasps, and a swift counter attacking move involving premature substitute Shaquille Gwengue finding Dajon Golding in the area, who fired beyond Adams to put Worthing 3-1 up in the 73rd minute.
East Grinstead continued to probe despite the setback, and with 10 minutes remaining, Theopanous poked the ball beyond Fagan to reduce Worthing’s lead back to one goal.
Four minutes later, substitute Daniel Thompson found himself unmarked in the penalty area to nod home and equalise for East Grinstead.
Meekums came close to a winner with a penalty shootout looming after a mazy run saw him fire low beyond Adams and narrowly past the post, before Theopanous carved out a magnificent chance to win the settle the tie in the Wasps’ favour, but his effort went wide leaving a penalty shootout to decide the victors.
Substitute Ollie Peace and Colbran were on target for Worthing, as were Sean Clohessey and Thompson for East Grinstead.
Armstrong then stepped up Worthing, but saw his effort saved by Adams, and Amoo fired beyond Fagan in response to put the Wasps ahead.
Goalscorer Golding netted the Reds’ next penalty, as did Danny Waldren for the hosts to mount the pressure.
Aguiar bravely took the instant sudden-death penalty for Worthing, but he fired his effort over the bar to send East Grinstead through to the next round of the competition.