The Scottish Premier League club looked to be on easy street after Dargo's breakaway goal seconds before the interval added to his seventh-minute opener.
Saints missed several chances to clinch the tie - and Dargo should have had a hat-trick - before a Mark Roberts' penalty in the 57th minute led to what had at one point looked like an unlikely fightback.
However, the part-timers ran out of steam at the end although Danny Lennon and his St Mirren players were glad to survive to take their place in tomorrow's draw.
It was an old-fashioned cup tie which evoked memories of a time gone by.
Ayr's traditional stadium with terracing on three sides had a healthy crowd of 5,997 for the visit of the Buddies.
The Honest Men, sitting second in the table, had beaten Hibernian in the last round to set up the tie with the Paisley men so the home fans harboured hopes of another cup upset.
But those hopes were deflated in the seventh minute when the diminutive Dargo fired St Mirren in to the lead.
The former Kilmarnock striker took a David Van Zanten pass at the edge of the Ayr box, spun past defender Martyn Campbell and drove a left-footed shot low past David Crawford.
As the visitors took a grip, defender Darren McGregor burst through the Somerset side and in to the box with impressive pace before slashing the ball wide of the target.
Then Paisley midfielder Paul McGowan cut in from the left before rifling a right-footed drive wide of Crawford's right-hand post.
St Mirren looked firmly in control and Dargo only just failed to get on the end of a measured dink over the United defence by McGowan.
In the 27th minute Ayr, playing with only skipper Roberts in attack, broke with confidence for the first time.
Ryan McCann and Stuart Bannigan linked up well down the left but the latter slammed the ball in to the side-netting from the byline when he ought to have posed more of a problem for the Buddies' defence.
Moments later Saints striker Michael Higdon was fortunate only to get a yellow card from referee Euan Norris when he went through Ayr left-back Eddie Malone in what seemed to be a continuation of an earlier verbal spat between the two players.
A mistake by Ayr's veteran defender Jim Lauchlan in the 40th minute allowed McGowan a chance 10 yards out but he snatched at the opportunity and the ball went wide of the far post.
In the final minute, as the home side pressed, midfielder Chris Smith headed a Scott McLaughlin header just over the bar before complaining that it had come off a defender's head but the home side had more to moan about with seconds remaining.
Dargo raced clear on to a Higdon pass, rounded Crawford and with Lauchlan and Campbell back in attendance, steadied himself before slotting in number two.
The loss of a second goal seconds from the break saw the part-timers slope off dejected at half time and they should have been punished again soon after the restart.
In the 49th minute Dargo should have secured the match ball when he latched onto a Higdon header to ghost past the United defence. His shot beat Crawford but slipped agonisingly past the far post.
Moments later Dargo was set up at the edge of the box by McGowan and again his drive went wide of the target.
It looked rather comfortable for the SPL side as the chances kept coming.
In the 56th minute McGowan's cross from the left was met by Higdon but his close-range header was well saved by Crawford.
The game should have been finished by this point but a minute later, after the visitors had robustly defended an Ayr attack at the expense of a corner, the home side were back in the cup tie.
McLaughlin's corner from the right struck a St Mirren hand in the box - it appeared to be McGregor - and after referee Norris pointed to the spot Roberts sent Buddies' keeper Paul Gallacher the wrong way from the spot.
United, for the first time in the afternoon, began taking the game to St Mirren.
Dargo was replaced by teenager Kenny McLean in the 75th minute - a substitution which did not go down well at all with the travelling support.
Five minutes later Ayr had half a chance to level when McLaughlin knocked the ball over the Paisley defence but Ross Robertson was stretching and could not get full contact on the ball, allowing Gallacher to save.
Ayr kept going valiantly but ran out of time before they could carve out a real chance to win themselves a replay.