Replying to Essex’s 168, the reigning champions made 385, with wicketkeeper Burgess following his 178 against Surrey with 170 off 217 balls.
Matt Lamb added 71 and pressure on Essex quickly told as they lost Nick Browne early in the second innings to reach stumps at nought for one.
At the Kia Oval, England all-rounder Sam Curran scored 44 not out in his first knock since recovering from a stress fracture of his back as Surrey replied with 204 for five to Somerset’s 337.
Ben Foakes hit 63, enjoying an 80-run fourth-wicket partnership with Ollie Pope, who contributed an attractive 47.
Peter Siddle struck an important blow for Somerset when he had Foakes caught at second slip from the last ball of the 59th over.
England hopeful Josh Bohannon scored his fourth first-class century as Lancashire totally dominated against Gloucestershire at Emirates Old Trafford.
The Red Rose closed on 289 for three and with a lead of 37 runs on a day when just two wickets fell.
Bohannon finished unbeaten on 142 alongside his skipper Dane Vilas (39) after opener Luke Wells had earlier made a well-constructed 59.
A blistering century from James Vince put Hampshire in charge of their match with Kent at Canterbury.
Hampshire were 337 for four at stumps, a lead of 32, after Vince made 111 from 118 balls – including 19 fours.
Kent’s Nathan Gilchrist took three for 71, but it was a day to forget for the hosts, who were bowled out for 305, with Keith Barker claiming six for 53.
Yorkshire seized control against Northamptonshire by building a 177-run advantage at Wantage Road.
Adam Lyth and Dimuth Karunaratne shared an opening partnership of 73 as the visitors built on their first-innings lead of 92 after Matthew Revis and Dom Bess shone with the ball to restrict the hosts to 204 all out.
Revis took three for 43, his best first-class performance, while Bess returned miserly figures of two for 31 from 22 overs.
In Division Two, Shan Masood became the first player in Derbyshire history to post back-to-back double centuries as Leicestershire’s bowlers were put to the sword.
The Pakistan international was eventually out for 219, having hit 26 fours and a six, as the visitors racked up 401 runs in the day to reach 437 for four and lead by 224.
Masood’s magnificent innings followed his 239 in last week’s draw against Sussex, while Wayne Madsen made 94 and Mattie McKiernan ended the day 63 not out.
Worcestershire club captain Brett D’Oliveira continued his prolific start to the campaign with an unbeaten 169 against Sussex at New Road.
Joe Leach picked up three wickets in seven balls late in the day to put Worcestershire in control as the visitors, despite an unbeaten 85 from Cheteshwar Pujara, closed on 169 for five.
All-rounder D’Oliveira batted for seven and three quarter hours in total and was the bedrock of his side’s 491 all out.
Unbeaten centuries from Ben Slater and Lyndon James provided Nottinghamshire with a dominant platform against Durham as they closed on 371 for three at Chester-le-Street.
Slater (164) and James (105) enjoyed an unbroken stand of 216 to help their side build a 141-run lead.
Matthew Potts claimed figures of three for 88 on a difficult day for the home side, who were dismissed for 230.
In Cardiff, Middlesex took a complete stranglehold against Glamorgan as the home side finished 104 for six in their second innings, still 110 runs behind.
An undefeated John Simpson hundred and a fifty from Toby Roland-Jones meant Middlesex made 336 all out to give them a 214-run first-innings lead.
Glamorgan bowler James Harris claimed four for 68 against his former county.