Sw$i$ndon spoiled Rovers' 125 year celebrations by taking all three points from the League One match at Prenton Park.
Tranmere fans got the chance to see around 40 former players on the pitch before kick off but the present day side paid for failing to take advantage of their early superiority.
Rovers had chances for Thomas-Moore, Edds, Curran and McLaren before two goals from Billy Paynter put the visitors in control. A third from McNamee looked like it had killed the game off but Shuker's header kept things interesting in the late stages until Simon Ferry added a fourth in injury time.
Caretaker Manager Les Parry made one change to the side that started at Hartlepool with Michael Ricketts returning from illness to take the place of Terry Gornell in attack.
Swindon's Billy Paynter had a chance in the second minute of the game, but his close range effort was scuffed straight at David Martin. At the other end a Ricketts header caused a moment of panic in the Swindon area and Ian Thomas-Moore almost got in front of visiting keeper David Lucas but the ball was turned away.
Ricketts did exceptionally well on five minutes to hold off a defender on the right wing and his curling cross was just beyond the outstretched leg of Gareth Edds at the far post.
On 11 minutes a McLaren corner was met by Goodison six yards out and his downward header had to be turned around the post by Lucas. From the corner that followed the ball fell to Curran in a central position and this time it was a volley that was saved by Lucas.
Curran showed his athleticism moments later as he raced back into his own area to clear the danger as Swindon quickly broke from Rovers' set piece.
A foul on John Welsh gave Rovers a free kick in a central position 25 yards out on 19 minutes, and Lucas was forced into another save as McLaren aimed his shot just inside the right hand post.
Ian Goodison did well to clear a cross from Jon-Paul McGovern on 24 minutes while Ricketts held the ball up well inside the Swindon area a minute later before laying off for Gareth Edds to smash a volley towards goal which was kept out by Lucas.
Michael Timlin fired high over the bar from outside the area on 33 minutes but the Robins' next chance brought the game's opening goal. A cross from the right from McGovern wasn't cleared by the defence and Billy Paynter nipped in front of Goodison to turn past Martin from close range.
Two minutes later appeals for a penalty for Rovers were turned down when Curran's run was blocked 12 yards out after good work from Ricketts and Thomas-Moore.
However, on 39 minutes the visitors went 2-0 up. Zoumana Bakayogo's attempted clearance deflected into the path of Kevin Amankwaah and his cross was curled into the path of Paynter who shot across Martin and into the net for his second of the afternoon.
Paynter had a chance for his hat-trick on 43 minutes but a bad touch as he raced through on goal allowed Goodison to get back and deny him a decent shooting opportunity.
A minute before half time the striker missed another chance, this time his hooked effort cleared by Broomes as the Robins took charge.
Goodison produced a wonderful tackle to deny Paynter a clear run through on goal on 48 minutes while a cross from Thomas-Moore flew just past the far post a minute later with Curran just unable to make contact.
Paytner - looking hugely dangerous every time he was on the ball - rattled Martin's left hand post on 49 minutes with a drive from outside the area and another effort had to be cleared off the line by Bakayogo.
Martin saved from Cuthbert on 54 minutes and was fortunate to see Obadeyi drag his shot wide a minute later.
Paul McLaren was replaced by Chris Shuker on 57 minutes and Rovers switched to 4-4-2 with Shuker lining up on the left wing and Curran on the right.
The visitors made it 3-0 on 65 minutes. Obadeyi raced clear down the left and his cross fell to Anthony McNamee who fired low into the left side of the net from 15 yards.
Gornell was introduced in place of Ricketts with just over 20 minutes remaining while Edds curled into the arms of Lucas as he looked to drag his side back into the game.
The home side got their goal on 78 minutes. A cross from the left was flicked on by Goodison and Shuker raced in at the far post to duck down and head into the net.
Moments later the officials failed to give what looked a clear handball on the edge of the Swindon area while a good run from Shuker ended when the winger lost his balance after turning a defender.
Goodison created a chance with an audacious short corner but the Jamaican was unable to find a team mate after racing into the area.
A shot from Welsh went over the bar as the game went into injury time and soon after Swindon scored their fourth when McGovern's short corner was expertly turned into the bottom corner from outside the area by Simon Ferry.
Tranmere: Martin, Logan, Broomes, Goodison, Bakayogo, Edds, Welsh, McLaren (Shuker 57), Curran, Ricketts (Gornell 68), Thomas-Moore. Substitutes not used: Daniels, Taylor, Cresswell, Mahon, Fraughan.
Swindon: Lucas, Cuthbert, Greer, McGovern, Timlin (Easton 77), McNamee, Amankwaah, Jean-Francois, Paynter (Peacock 87), Obadeyi (Austin 84), Ferry. Substitutes not used: Hutchinson, Smith, Morrison, Revell.
Referee: Tony Bates
Yellow cards: Welsh (36), Thomas-Moore (60), Jean-Francois (83)