The Swans were pegged back twice by Newcastle as they were forced to share the spoils at the Liberty.Wilfried Bony had put the Swans ahead with his first goal of the season after 17 minutes following some neat link-up play with Gylfi Sigurdsson.
But despite their dominance, the Swans couldn't add to their lead and they went into the break on level terms as Papiss Cisse touched home Gabriel Obertan's low cross.
Garry Monk's side were back in the lead five minutes after the restart as Sigurdsson slipped through for Routledge, and the former Newcastle winger chipped over the onrushing Tim Krul.
Krul then did superbly well to prevent Bony from adding a third, and it paid off for the visitors as Cisse's neat finish from sub Sammy Ameobi's cross made it 2-2.
Marvin Emnes, on for Sigurdsson, almost restored the Swans' lead, but again he was denied by Krul.
It meant a point for both sides, with the Swans going into the international break with 11 points from seven games.
The Swans made two changes to their starting line-up.
With Angel Rangel serving his one-match suspension following his dismissal for two bookings at Sunderland, Jazz Richards made his first Premier League start of the season.
The other change saw Wilfried Bony replace Bafe Gomis up front, while, on the bench, 19-year-old Liam Shephard was drafted in from the club's Under-21 set-up as he provided cover at full-back in what was his first inclusion in the first-team matchday squad.
Dwight Tiendalli, Jordi Amat and Leon Britton remained on the injury list for the hosts.
Lukasz Fabianski, Jazz Richards, Federico Fernandez, Ashley Williams (capt), Neil Taylor, Ki Sung-Yueng, Jonjo Shelvey, Gylfi Sigurdsson (Marvin Emnes, 84), Nathan Dyer (Jefferson Montero, 84), Wayne Routledge, Wilfried Bony.
Gerhard Tremmel, Liam Shephard, Kyle Bartley, Tom Carroll, Bafe Gomis.
Tim Krul, Fabricio Coloccini (capt), Mike Williamson, Moussa Sissoko, Papiss Cisse (Ayoze Perez, 84), Yoan Gouffran (Sammy Ameobi, 58), Jack Colback (Emmanuel Riviere, 69), Daryl Janmaat, Cheik Tiote, Gabriel Obertan, Paul Dummett.
Rob Elliott, Vernon Anita, Massadio Haidara, Remy Cabella.
The Swans settled quickly into the match and enjoyed the majority of possession and territory.
Although they didn't create many opportunities early on, the likes of Jonjo Shelvey, Gylfi Sigurdsson and Wilfried Bony were looking lively.
It took until the 17th minute for the Swans to carve open the Newcastle defence for the first time, and they took their chance.
After Gabriel Obertan had fired low and wide for the visitors, the Swans moved upfield and a neat one-two between Bony and Sigurdsson following Ki's initial pass sent the striker in on goal. And the Ivorian found the bottom corner with a neat right-footed finish for his first goal of the season.
Garry Monk's side went in search of another goal, and Ki's chipped pass almost sent Sigurdsson in on goal but the Icelandic midfielder didn't manage to get onto the end of it.
The whole of the Liberty Stadium then burst into applause following a good luck message appearing on the big screens for Newcastle midfielder Jonas Gutierrez, who is battling cancer.
Shelvey then curled a 20-yard effort over the crossbar before his chipped pass into the box again failed to find the onrushing Sigurdsson.
Sigurdsson was at the centre of most Swans attacks, and his inswinging free-kick found Ki unmarked, but his header from eight yards bounced wide before he was ruled offside.
The Swans were probing heavily for a second goal, and Bony was denied by a superb save from Tim Krul as the Dutchman scooped low to palm his volley wide for a corner.
Newcastle, though, went close to levelling after Mike Williamson - under pressure from Jazz Richards - headed over the crossbar just eight yards out from a Yoan Gouffran corner.
And the visitors went one better soon after as Obertan's low cross was sidefooted home by Papiss Cisse just six yards out with two minutes left of the half.
It failed to rattle the home side though, and Ki and Bony were twice denied in quick succession before Wayne Routledge's fierce volley just dipped over the crossbar.
Half-time: Swans 1 Newcastle 1
The frustration of conceding the late goal in the first half was soon forgotten as the Swans regained the lead in the 50th minute.
Sigurdsson was again the provider as the midfielder won possession back off Sissoko before slipping through a penetrative pass between Coloccini and Williamson which Routledge latched onto before dinking his 12-yard effort over the advancing Krul.
Alan Pardew made his first change just before the hour as Sammy Ameobi replaced Gouffran, but it was the hosts who continued to work towards another goal.
And Bony went close as some fine link up with Sigurdsson, and a back-heeled pass from the latter, set up the striker but his shot from an awkward angle failed to test Krul.
Pardew then introduced Emmanuel Riviere in place of Jack Colback as visitors gave themselves more attacking options.
But it was Krul who again came to the visitors' rescue as Nathan Dyer's low cross found Bony, and his first-time shot was turned around the post by the Dutch keeper's fine reflex save.
And it paid dividends for Newcastle as they were level minutes later.
Sub Ameobi's superb cross found Cisse, and the striker guided home from close range in the 75th minute.
Both sides made changes six minutes to go as Ayoze Perez replaced Cisse while Jefferson Montero and Marvin Emnes came on for Dyer and Sigurdsson.
And Emnes was straight into the action as he latched onto Bony's through ball, but Krul's feet denied the Dutch winger's low effort.
Then, in added on time, Daryl Janmaat's swerving effort was well saved by Fabianski, who was under pressure from Perez.
But that was the end of the action as both sides claimed a point.