Bangladesh had looked in danger of becoming only the fifth side to lose a Test in two days since World War Two but clung on to close Saturday on 297 for eight, still needing 46 to make the home side bat again.
Hoggard, who had taken three wickets for two runs late on Saturday, removed Tapash Baisya in the fourth over of a drab morning as the tailender edged behind.
Mashrafe Mortaza followed in similar fashion five balls later, Marcus Trescothick holding the catch at first slip at the second attempt, as Bangladesh were bowled out for 316.
Aftab Ahmed, the last recognised batsman, was left stranded on a run-a-ball 82 not out, hitting one six and 13 fours. England needed only 26 balls to complete their win on Saturday.
It is the first time England have won five successive Test series since 1969-71. It was also the side's 14th win in 18 Test starts, set against a single defeat. Bangladesh have won just one of their 38 Tests.
Bangladesh's final innings of the series easily surpassed their three previous efforts -- 108 and 159 at Lord's, followed by their first innings of 104 at The Riverside.
England's 447 for three declared was built on Trescothick's 151 and Ian Bell's 162 not out, his maiden century in his third test.
The two teams next play in a one-day triangular series also involving Australia. England take on Bangladesh in the first match at The Oval on June 16.
Scoreboard
Bangladesh 1st innings: 104 (S. Harmison 5-38)
England 1st innings: 447 for 3 decl (I.Bell 162 n.o., M.Trescothick 151, G.Thorpe 70 n.o.)
Bangladesh 2nd innings (Overnight: 297-8)
J. Omar c G. Jones b Harmison 71
N. Iqbal c G. Jones b Flintoff 15
R. Saleh c Strauss b Flintoff 7
M. Ashraful c Hoggard b Batty 12
H. Bashar lbw b Flintoff 63
K. Mashud lbw b Hoggard 25
A. Ahmed not out 82
M. Rafique b Hoggard 2
A. Hossain c Thorpe b Hoggard 0
T. Baisya c G. Jones b Hoggard 18
M. Mortaza c Trescothick b Hoggard 0
Extras (lb-6 nb-14 w-1) 21
Total (all out, 72.5 overs) 316
Fall of wickets: 1-50, 2-75, 3-101, 4-125, 5-195, 6-235, 7-245, 8-251, 9-311, 10-316
Bowling
Hoggard 15.5 - 3 - 73 - 5 (nb-8)
Harmison 17 - 1 - 86 - 1 (nb-3)
S. Jones 10 - 1 - 49 - 0
Flintoff 15 - 2 - 58 - 3 (w-1)
Batty 15 - 2 - 44 - 1