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Gardner's 85th-minute volley extended the lead Stephane Sessegnon's fine 14th-minute finish had given the Black Cats and killed off any hope of a comeback by the Swans.
The former Aston Villa and Birmingham player, who was introduced as a 78th-minute replacement for David Vaughan, has been linked with a move back to the midlands this month after struggling to settle on Wearside following his £5million summer move.
However, O'Neill is ever more keen for him to stay.
He said: "We want him to stay at the football club. At this moment, there has been no concrete offer.
"There is interest from other Premier League teams in and around that area, but there has been nothing concrete and we wouldn't want him to be leaving.
"I have not thought about an immediate replacement at this moment, so time is running out. We want him to stay.
"Obviously at home, he is very family-orientated and he has never really been away before.
"But it is England and you can get from one place to another. It isn't impossible these days.
"It's not as if to say he is playing in Saudi Arabia."
Gardner's late intervention, which helped the Black Cats climb into 10th place in the Premier League table, was just what his side needed with Swansea stubbornly refusing to throw in the towel.
The home side had taken a 14th-minute lead when Sessegnon exchanged passes with James McClean before firing expertly past keeper Michel Vorm into the top corner.
But far from being wounded by the early reverse, the visitors took charge and dominated the rest of the half, although without ever testing keeper Simon Mignolet.
O'Neill was clearly unhappy with his side's display, and whatever he said at the break, they responded, although they had to wait until the 85th minute for Gardner to thump a dipping volley into the far corner.
O'Neill said: "It was a difficult afternoon for us. Swansea are a very, very fine footballing side who came here with plenty of confidence, not just because of their win against Arsenal.
"They have been playing excellently and it didn't matter whether we were the next opponents or not, they were going to play very well.
"They caused us problems in terms of possession, but the win for us was the most important issue.
"Two great goals decided the game, but it was a tough afternoon."
Swansea boss Brendan Rodgers had no complaints about his team's performance.
He said: "I thought it was a very, very harsh scoreline. Obviously, in terms of the result, it was disappointing, but the performance level was so high.
"I thought we were fantastic today. We created chances and maybe at times our final ball let us down a wee bit, but I can have no complaints.
"It was hard to take when you play ever so well, but you take your medicine and move on."
Rodgers was relaxed too over a Sebastian Larsson tackle which infuriated Scott Sinclair, but went unpunished by referee Chris Foy.
He said: "If I am being totally honest, it's a tackle that happens in the game and I have got no problem with it. I didn't think it was dangerous.
"But in the modern game, you have seen players sent off for that and in particular, Chris Foy has sent players off for that, so maybe he has just reined himself in a bit and taken a different stance on it."
Meanwhile, Sunderland striker Nicklas Bendtner will undergo a scan on an eye injury suffered in a challenge with Angel Rangel, with O'Neill hoping there is no lasting damage.
