The West Indies resumed the day on 93/3 and looking to stretch a slender 22 run lead into a defendable target.
Hope lasted just four balls after lunch, though, before Anderson found late movement down the slope to have him caught-behind and then doubled up by snaking through Devendra Bishoo's defences for a second-ball duck.
Anderson's exploits - Roach was his 39th victim of the Test summer - left England needing 107 runs for victory, a tally they eased to thanks to an unbroken second-wicket stand of 72 between Ashes hopefuls Tom Westley (44no) and Mark Stoneman (40no).
Play started in bright sunshine, a marked contrast to the grey skies that had encircled Lord's during the first two days.
Earlier, Anderson had set the tone by removing Roston Chase in the first over day, as he got the West Indian batsman to edge a delivery just outside the line off stump through to Jonny Bairstow.
Broad thought he had atoned for dropping Blackwood by pinning the right-hander lbw for four, only to see Blackwood's review prove successful with UltraEdge indicating a faint sliver of bat.
Chase was out for his overnight three and West Indies were 94 for four.
Anderson, who became the first Englishman, third seamer and sixth bowler to 500 Test wickets on day two at headquarters, scooped his 24th Test five-for and fifth at Lord's as Windies were shot out for 177.
It all made little difference when Blackwood was dismissed for five, caught behind off Anderson after pushing well away from his body.
The 23-year-old became the first player to score two centuries in consecutive matches at Headingley in the second Test and was again the man the West Indies looked to build their counterattack around.
Jermaine Blackwood (5) went similarly cheaply to Anderson from a short delivery outside off and, after Shane Dowrich (14) lobbed Toby Roland-Jones to short mid-on, Anderson ended Hope's admirable stay at the crease by drawing the outside edge with a stunning display of swing bowling.
Hope remained patient, waiting until lunch was in the offing before he cover-drove Ben Stokes for his first four of the day - from the 54th ball he had faced.
Hope, who also made a stylish 39 in hard conditions in the first innings at Lord's, then drove Roland-Jones past backward point for a third four in five balls.
Skipper Jason Holder would help Shai Hope pass fifty and get to lunch on 153/6 but he did receive a slice of luck being dropped off the bowing of Joe Root off the final ball of the session.
But Alastair Cook at gully was unable to react in time to hold what would have been a spectacular catch.