Sibley was the first England wicket to fall after a 114-run partnership with Burns, trapped leg before wicket by Windies captain Jason Holder.
Left-handed Broad counter-attacked, pulling Roach for six and cut Holder for four.
By that point, Holder and Dowrich had moved West Indies from their overnight 137-6 past the follow-on mark of 170 with relative ease, albeit if Holder was held by a flying Ollie Pope off a marginal Chris Woakes no-ball.
England made two changes to the side that won the second Test by 113 runs at the Manchester ground on Monday to level the three-match series at 1-1.
The collapse brought Broad to the middle and the left-hander hit the Windies' bowlers to all parts of Old Trafford, reaching his half-century in 33 balls - putting him tied for third place in the all-time list of England's fastest test fifties.
His two wickets in the evening took him to within one wicket of becoming only the second ever Englishman, after Jimmy Anderson, to take 500 Test wickets. Captain Jason Holder took a painful blow to the thumb while fielding at second slip, while tired opening bowler Shannon Gabriel could only manage five overs.
The introduction of Holder only accelerated things, with Broad blazing seven boundaries off the skipper as he dropped too short and wide in a painful spell.
Botham, 64, who scored more than 5,000 runs and took 383 wickets in 102 Tests, said: "Viv was the finest batsman I ever played against".
The fun eventually stopped when he pinged a Roston Chase full toss to Jermaine Blackwood on the ropes, with Anderson last man out for 11 leaving Dom Bess stranded on 18 not out.
Rahkeem Cornwall takes a pearler of a catch against England.
He then took a wicket with his fourth ball to have West Indies dangerman Kraigg Brathwaite caught at slip by England captain Joe Root and, at tea, the visitors were 59-3.
"You can't compare anyone in India at the moment with Ben Stokes".
Jofra Archer struck with a dynamite delivery, the ball arrowing towards the ribs of John Campbell, who had looked good in reaching 32 but could only paddle away a snorter of a ball to Rory Burns at gully. Archer endured a luckless last burst and was halted mid-over by the fading light.
"But I think it's more about doing a job for your team which I'm very aware of and if you are not quite performing as well as you want, you realise you're not doing your job as well as you should be for your team". Click the link in the orange box below for details.
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