The winger scored a quickfire first-half double at the Madejski Stadium to put the Welsh outfit in control, before Nick Blackman's wonder-goal ensured Reading trailed by just one at the break.
Undeterred by the setback, Swansea restored their two-goal cushion courtesy of a neat finish from Bafetimbi Gomis in the second half.
The Premier League side had named a youthful team in a 3-1 defeat to Bournemouth on Friday, but named a more familiar-looking line-up on this occasion.
And Swansea spurned an early chance to open the scoring, Wilfried Bony smashing a penalty against the crossbar in the 14th minute after Sean Morrison had handled Dwight Tiendalli's cross.
The Ivory Coast international atoned for his error with the assist for Routledge's opener in the 23rd minute, threading through to the winger who rounded the goalkeeper and tapped home.
It took just two minutes for Swansea to double the lead. A swift break saw the Welsh outfit outnumber the hosts' defence and former Reading man Gylfi Sigurdsson set up Routledge to convert into an empty net.
Reading came back into the game shortly before the break when Blackman let fly with a thunderous 30-yard drive that left Lukasz Fabianski helpless.
Gomis then struck the post from 20 yards, before capitalising on a mix-up between Morrison and Michael Hector to race clear and finish with a neat chip over Adam Federici in the 65th minute.
Substitute Jefferson Montero, who replaced Routledge, almost made an instant impact with a strong shot from the left that Federici saved well.
Swansea applied more pressure in the closing stages but failed to add a fourth goal.