Tyler Walton scored a double – including a 92nd minute equaliser – to send the game in extra-time although Swindon were down to ten-men.
It came down to a shoot-out with Everton loanee Crellin saving two penalties - and one in the 90 minutes - to send Stanley through 4-1 on penalties.
Defender Faz Rawson was suspended for this game after his sending off at AFC Wimbledon.
It meant Donald Love started in the backline with Connor O’Brien coming in to start.
The first chances fell to Swindon with Harry Smith and George Cox unable to find the target.
Stanley opened the scoring on 17 minutes on their first real attack. Robins keeper Jack Bycroft pushed away a Walton strike, the ball was loose in the box, with Liam Coyle and Josh Woods having a go, before Walton got the final touch to force the ball over the line.
There were chances at both ends with a Swindon header over and Coyle unable to hit the target.
Swindon were reduced to 10-men on 41 minutes when Danny Butterworth saw red for a high tackle on Coyle in midfield.
The Robins however kept pressing and equalised in the second minute of time added-on when Grant Hall headed home at the far post.
After the re-start, the Reds had a great chance to go ahead again on 52 minutes with Josh Woods stretching onto a ball in the area but he just couldn’t connect to turn it home.
Dara Costelloe then had an opportunity from close range on 55 minutes but the [Burnley](/news/burnley/) loan forward put the ball over the crossbar.
Swindon were awarded a penalty on 63 minutes for a foul by Coyle on Cox but Crellin was superb in keeping out Smith’s fierce spot kick.
Costelloe had an effort saved by Bycroft before Swindon re-took the lead on 70 minutes with a solo effort from substitute Joel Cotterill who ran to the edge of the box and curled the ball home. Swindon could have got a third, Crellin superbly denying Smith who was clean through on 87 minutes while, in the 89th minute, Bycroft managed to keep out a flick from Walton.
However, in the first minute of added time, Nelson Khumbeni’s strike was pushed out by Bycroft and Walton was there for his second of the game to take the tie into extra-time.
Costelloe fired across the goal and wide while Stanley threw their bodies in front of a number of shots in the only real action in the first half of extra-time.
In the second half, Walton had a shot cleared off the line in the opening stages while, towards the end, a Jimmy Knowles header was over the bar.
Crellin parried a Drinan shot away in the final minute of added time and that was the last of the action as the tie went to penalties.
Anjola Popoola scored the first and Crellin saved Drinan’s effort to give Stanley a 1-0 lead.
Knowles fired home but Crellin was there again to deny Cotterill and Staney led 2-0.
Ben Woods found the back of the net as did Will Wright to make it 3-1.
Khumbeni scored the winning spot kick for 4-1 to put Stanley in the hat in the third round draw which will take place this evening at 7pm.
Accrington Stanley: Crellin, Aljofree, Love, Awe, O’Brien (Hunter), Coyle (Conneely), Khumbeni, B Woods, Costelloe (Popoola), Walton (Knowles), J Woods (Henderson)
Subs: Kelly, Smith, Sohna, Trickett
Crowd: 1534 (355 from Swindon)