Exeter led 2-0 at half time over the side who finished bottom of League One last season, but were pegged back to 2-2 through goals from Marc Laird and Matt Dolan before substitute Hoskins clinched it.
Another relegated side, Notts County, started League Two life with a 2-0 win over Stevenage thanks to goals from Curtis Thompson and a late Mawouna Amevor strike, before Amevor was sent off for an injury-time clash with Tom Hitchcock, who was also dismissed.
Cambridge United will end the opening weekend of the season at the top of the table after a comfortable 3-0 victory over Newport County, while a double from Kyle Bennett helped Portsmouth overcome Dagenham & Redbridge by the same scoreline.
Stephane Zubar's own goal rounded off a miserable day for York City, who lost 3-0 at Wycombe Wanderers, and there were 2-0 wins for Plymouth Argyle away at AFC Wimbledon and Hartlepool at home to Morecambe.
Results in full (half-time scores in brackets): Accrington Stanley 1-1 Luton Town (0-0), AFC Wimbledon 0-2 Plymouth Argyle (0-1), Bristol Rovers 0-1 Northampton Town (0-0), Cambridge United 3-0 Newport County (1-0), Exeter City 3-2 Yeovil Town (2-0), Hartlepool United 2-0 Morecambe (1-0), Leyton Orient 2-0 Barnet (0-0), Mansfield Town 1-1 Carlisle United (1-1), Oxford United 1-1 Crawley Town (0-0), Portsmouth 3-0 Dagenham & Redbridge (0-0), Stevenage 0-2 Notts County (0-1), Wycombe Wanderers 3-0 York City (2-0)