Wins over Bury and Accrington Stanley, along with a goalless draw against Carlisle, had helped put Cheltenham among the early front runners in the fourth tier.
However, they endured a difficult opening 45 minutes at Prenton Park as Eliot Richards and Marc Laird gave the hosts a two-goal cushion at the break.
However, Rob Edwards' men had picked up just four points from their opening three games and meekly surrendered their advantage in the second half.
Three goals in the space of just six minutes from Matt Richards, Koby Arthur and Jordan Wynter helped the visitors return home with maximum points as Mark Yates' charges stayed within two points of leaders Morecambe.
Paul Mullin's brace was enough for a 2-1 win over Cambridge United, and Morecambe are the only team still with a 100 per cent record in the Football League this term.
Portsmouth sit second after Craig Westcarr's goal 10 minutes from time helped them see off winless Oxford United while Burton Albion's winning start ended at Newport County.
Chris Zebroski's first-half opener looked to have condemned the visitors to defeat but John Mousinho levelled six minutes after the break to keep Burton unbeaten.
The draw ensures Newport are off the mark with their first point of the season while Accrington were able to do the same against Luton Town.
However, James Beattie will likely wonder whether his side could have taken maximum points, having led 2-0 after 16 minutes through James Gray and Luke Joyce.
The Football League newcomers showed their spirit to draw level before the break courtesy of Luke Rooney and Jake Howells in an entertaining affair at The Crown Ground.
York City drew for a fourth consecutive game as Keith Lowe equalised late on against Exeter City, while Emile Sinclair's 90th-minute leveller ensured a share of the spoils for Northampton Town against 10-man Shrewsbury Town.
A debut goal from Charlie Wyke helped Hartlepool United see off his former club AFC Wimbledon 2-1 as Wycombe Wanderers and Dagenham and Redbridge overcame Stevenage and Mansfield Town respectively.
And a goal apiece from Antony Sweeney and Conor Clifford ensured the points were shared between Carlisle United and Southend United at Brunton Park, while Bury were too strong for Plymouth Argyle in their 2-1 win.
