Swansea showed four changes from the side that earned a draw at Doncaster with Leon Britton, Darren Pratley, Nathan Dyer and Stephen Dobbie all making welcome returns to the starting line-up.
Rangers kept the side that drew 1-1 at Newcastle.
A low cross from Mikele Liegertwood in the third minute was turned behind for a corner by Ashley Williams. The Swans failed to clear Ben Watson's flag kick and the ball ran out to Akos Buzsaky who drove his shot inches over the bar.
Rangers kept up the pressure and Wayne Routledge fired just wide after running clear of the home defence.
The visitors were stroking the ball about with some purpose but the Swans more than matched them particularly when Nathan Dyer had the ball down the right flank.
Midway through the half, Dobbie met a cross from Angel Rangel, but Kaspars Gorkss cleared for a corner just as Swansea's striker was about to shoot.
Just before the interval, Pratley sent a low shot just wide of Radek Cerny's far post and then a centre from Dyer went right across the Rangers goalmouth but Dobbie was unable to get a touch.
Four minutes into the second half Dorus de Vries, making his 150th first-team appearance, pulled off a tremendous save to deny Jay Simpson from point-blank range and then Gorkss headed over when well placed.
In the 53rd minute, Rangers were down to ten men when skipper Martin Rowlands was shown a second yellow card for a foul on Dyer after being booked in the 36th minute for fouling Dobbie.
Swansea immediately piled on the pressure and Pratley was denied after meeting a cross from Dyer.
Lee Trundle and Craig Beattie went on as the Swans looked for victory, but although their approach play was good, the final pass let them down until the 74th minute when Gower beat Damion Stewart to an Angel Rangel pass at the near post to lash home his first goal for the club.
Rangers had their second player dismissed in the 77th minute when Ben Watson was yellow-carded for handball after a first yellow card five minutes earlier.
A quick break down the right by Dyer made the game safe for the Swans. He made rapid ground before crossing to the unmarked Trundle, who sidefooted home.
It was all Swansea in the closing minutes but they were unable to add to their score.