In the first ever league meeting between the sides, Adam Le Fondre's penalty 13 minutes from time handed the Royals their first Sky Bet Championship win since the opening day - and condemned the Glovers to a fifth consecutive defeat in all competitions.
Yeovil boss Gary Johnson made two changes to the team that started in Tuesday's controversial Capital One Cup defeat against Birmingham City, with Republic of Ireland international Paddy Madden again missing out through injury.
The Glovers, who went into the game having not netted in the league since Ed Upson's late winner at Millwall on the opening day, started brightly and created a host of glorious opportunities in the opening period.
Defender Byron Webster - who was at the centre of Tuesday night's drama at Huish Park - headed narrowly over from a corner before midfield talisman Upson's driven effort was well saved by goalkeeper Alex McCarthy.
Meanwhile, the visitors, who were also looking for their first Championship win since their opening win over Ipswich Town, toiled in the Somerset sun and struggled to create any clear-cut chances.
However, the Glovers grew in confidence as the half went on and should have gone into the break ahead, with Joe Ralls, James Hayter and captain Jamie McAllister all spurning good openings.
Adkins swapped Gareth McCleary for Danny Williams in a surprise first-half substitution and the newly-introduced midfielder came closest for the Royals before the break when he volleyed former Real Madrid and Everton winger Royston Drenthe's deep cross wide of the post.
Yeovil again dictated proceedings at the start of the second half, with on-loan striker Andy Williams coming inches away from breaking the deadlock after hitting a curling effort narrowly wide.
Williams, who made his first start since arriving at Huish Park from Swindon Town, continued to stretch the Berkshire side's defence as the game became an open affair after the hour mark, but both goalkeepers remained relatively untested.
With 20 minutes remaining, Yeovil once again came close to snatching a deserved lead after the impressive Upson hit a low right-footed shot wide of McCarthy's post.
However, Huish Park once again became the centre of controversy on 77 minutes after referee Trevor Kettle awarded the Royals a penalty after Luke Ayling was adjudged to have brought down Le Fondre.
The striker picked himself up and slammed past goalkeeper Wayne Hennessey to net his third league goal of the season and earn the visitors the valuable three points.