The Scot finally made the breakthrough with a left-footed drive that meant his side leapfrogged Crystal Palace back to the summit and took their places in the Cardiff record books.
In the second period Gary Madine almost fired Dave Jones' men into the lead against his manager's former employers, only to be denied by the reactions of David Marshall.
Just when it looked like Wednesday would end the only 100 per cent home record in English league football, Conway intervened with his first goal of the season.
The Bluebirds have made a habit of scoring very early on in recent games and if Andrew Taylor's first-time effort had been a few inches lower from Craig Bellamy's smart pull-back after just 44 seconds then they would have done it once again. The Welshman's pace left Chris O'Grady for dead before he found the City left-back.
Taylor and Bellamy then swapped roles to create the Bluebirds' next opening with Bellamy missing the target with a header after Peter Whittingham's inch-perfect through-ball sent Taylor scampering down the left.
Jordon Mutch, who celebrated his 21st birthday with his first start since September, was next to get in on the act with a fizzing effort which was almost headed past his own keeper by Miguel Llera.
Cardiff were beginning to up the ante against a side 23 points below them and four minutes later Mutch tried his luck again, his powerful drive from distance parried by Owls goalkeeper Chris Kirkland before the ball was hacked to safety by Llera.
Despite their hosts' midfield dictating proceedings this Wednesday side showed little resemblance to the one comprehensively thrashed at home by Watford on Tuesday as they saw out the rest of the first half without any real difficulty.
Having found the back of the net in their last 17 Championship matches it seemed only a matter of time before Malky Mackay's men made the breakthrough.
Bellamy fired into the midriff of Kirkland and the former Coventry goalkeeper did well to hold onto Mark Hudson's flick from Bellamy's cross but Wednesday were proving a tough nut to crack.
Wednesday did switch off at a free-kick allowing Bellamy to cross for Conway to fire over the bar but a minute later only Marshall denied them a shock opener. Michail Antonio slipped the first-half substitute through and his low drive from the right-hand side of the box was well saved by the Scot's legs.
Then history was made with the next real chance in the 80th minute, substitute Joe Mason nodding the ball back into the path of Conway just outside the box. The former Dundee United winger controlled the ball on his right knee before firing a left-footed shot into Kirkland's bottom right-hand corner to spark wild celebrations in South Wales.