The pair shared a partnership of 71 in 13 overs that took West Indies to 457 all out, 41 runs ahead of England's first-innings 416, on the third day of the second Test.
This was West Indies' highest Test total since their 484-7 declared against Bangladesh at Kingstown in September 2014.
Da Silva was left on 82 not out after Shamar Joseph was dismissed for 33 off just 27 balls, including five fours and two spectacular sixes.
It had looked as if Saturday morning would belong to England paceman Chris Woakes, who took two wickets in two balls to leave West Indies 386-9 -- still 30 runs behind.
But Shamar Joseph pulled fast bowler Gus Atkinson for six to level the scores and, two balls later, hoisted him behind square with another six that knocked tiles off the roof of the Larwood and Voce Tavern.
West Indies' performance in Nottingham, which on Friday featured Kavem Hodge's maiden Test century, was all the more creditable given they had been dismissed for just 121 and 136 during an innings and 114-run defeat by England at Lord's.
That left them 1-0 down in this three-match series.
West Indies resumed on 351-5, 65 runs behind, after Hodge (120) and Alick Athanaze (82) shared a stand of 175 that revived them from 84-3.
England, in their first home Test since 2012 without retired duo James Anderson or Stuart Broad -- who between them took 1,308 wickets at this level -- struggled on a hot day.
An overcast Saturday morning promised to aid England's pacemen, armed with the new ball, as Jason Holder (23 not out) and Da Silva (32 not out) looked to narrow the deficit.
Holder was out for 27, caught behind off Woakes.
The all-rounder later had Alzarri Joseph caught behind and next ball Woakes bowled Jayden Seales before Shamar Joseph survived the hat-trick.
But Da Silva completed an 87-ball 50 in spectacular style when he launched Mark Wood high over extra-cover for six.
And the wicketkeeper then took West Indies past 400 with another impressive six, an extravagant uppercut off Wood.
England captain Ben Stokes turned to Joe Root to break the stand but Da Silva clubbed the part-time spinner for three fours and a six off successive deliveries.
Wood eventually ended the run spree when Shamar Joseph chipped him to mid-on.
