With just four minutes on the clock, Kuqi produced perhaps the miss of the season.
Ipswich took the lead when Kuqi found the target for the 19th time this season.
Joe Royle's side had the perfect start when Shefki Kuqi put them ahead after just four minutes.
Darren Bent and Shefki Kuqi were both prolific for Ipswich last season before joining Premiership clubs in the close season.
Danny Maddix, Gerald Sibon and Lloyd Owusu all went close before Shefki Kuqi missed from six yards with the goal at his mercy.
And a minute from time, Kuqi almost grabbed an equaliser for the visitors but his header from a Leon Knight corner went wide.
Shefki Kuqi got in front of his marker to flick it on and Paul McLaren thumped a header against the outside of the post.
Kuqi then made amends with his fifth goal of the season, throwing himself at Gower's cross and scuffing his header past a despairing Steele.
Playing staff also laid-off included Brian Kerr, Mickael Antoine-Curier, Eric Paton, Dominic Shimmin, Njazi Kuqi, Paul McHale, Scott Fox, Colin McMenamin and Charlie Grant.
Luke Steele parried a Darren Pratley header into Ashley Williams for the opener, then Shefki Kuqi glanced in a diving header off Mark Gower's cross.
Ipswich boss Joe Royle sees the former Chelsea trainee as the ideal replacement for top scorer Shefki Kuqi, who is set to leave Portman Road.
BBC: Ipswich have won the race for Swindon striker Sam Parkin.
Beresford, due to start talks about a new extended deal early next week, produced stunning saves to deny Kuqi and Sibon twice as Burnley held on to the win.
The on-loan Chelsea forward laid it off for Kuqi but, instead of shooting, the former Stockport striker took a touch and Davenport got back to take it off his toe.
Kuqi's height was causing the home defence all sorts of trouble, but it was on the ground the Finland international should have put his side ahead on the half hour.
He found Shefki Kuqi on the edge of the box and the Finn chested it down before turning and unleashing a low shot which Joe Lewis turned around the post.
Both keepers were soon in action, Wednesday's Kevin Pressman easily dealing with a 25-yard shot from Darren Wrack and his opposite number James Walker saving with similar ease from Shefki Kuqi.
Manager Paulo Sousa shrugged off criticism of his repeated tinkering with the team to make six changes, Federico Bessone, Albert Serran, Kuqi, Cedric van der Gun, Leon Britton and Gower all recalled.
Kuqi has turned his fortunes around at Selhurst Park after being placed on the transfer list last year by Warnock for making an offensive gesture at Palace fans, who cheered his substitution following a poor performance in the 2-0 defeat against Wolves.
Kuqi missed a sitter from 12 yards as the game entered the closing stages before finally finding the net at the third attempt, rounding off a successful evening for Swans boss Rodgers in what was his first competitive game at the Liberty Stadium.
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