England stand within a win of World Cup qualification after debutant Andros Townsend inspired a nervy win against Montenegro at Wembley.
The Tottenham Hotspur winger helped make England's first as he set up Danny Welbeck, whose shot eventually fell to Rooney, before Boskovic diverted the ball into his own net after the hour.
Dejan Damjanovic pulled one back for the away side but Townsend's superb strike from 25 yards, on his international debut, crowned the perfect evening for Hodgson's men who netted a fourth in stoppage time through Sturridge's penalty.
Despite Ukraine's win over Poland, England stand three points away from automatic qualification for next year's World Cup but are assured of at least a play-off place after as convincing a win during Hodgson's tenure as manager.
Knowing anything less than a win would likely condemn England to at best a play-off position in search of passage to Brazil and the possibility of not reaching a second major finals in three attempts, Hodgson chose an attack-minded team as his side went in pursuit of their first win of the group phase against a rival for qualification.
Townsend, winning his first international cap, and Welbeck flanked Rooney and Sturridge with Jack Wilshere making way for the experienced central pairing of Frank Lampard and Steven Gerrard.
Opposite number Branko Brnovic was in contrast short of options as his side went in search of a win that would put Montenegro on the brink of the top two, with Mirko Vucinic among six first-choice players nursing injuries.
A goal shy England thus far in the group phase began on the front foot, Rooney having his curled shot deflected wide of Vukasin Poleksic's goal before Sturridge's flighted effort went harmlessly over.
The fleet-footed Townsend was already causing Montenegrin defenders all kinds of problems while on the left Leighton Baines was providing equal thrust and when he slalomed away from Savo Pavicevic, Welbeck was denied by Poleksic from a tight angle.
Montenegro were showing little in the way of attacking threat and Joe Hart dealt effortlessly with Stevan Jovetic's shot when the Manchester City man broke in behind the England midfield.
Gerrard then forced Debrecen stopper Poleksic, Montenegro's third choice goalkeeper following injury to Mladen Bozovic, into turning his effort behind as the hosts rallied.
And Rooney should have handed England a half-time lead when Townsend's energetic run ended with his drive being parried, only for the Manchester United forward to hit the post from a tight angle.
Sturridge flicked wide after Rooney's centre on the stroke of half-time as the missed chances began to pile up for England and by the time they emerged for the second half, they knew Ukraine had leapfrogged them in the group with victory over Poland.
And they began seemingly realising the significance and duly opened the scoring three minutes after the break when Townsend's run and cross found Welbeck and though the United striker saw Poleksic keep him out, Rooney reacted first from six yards.
Rooney's sixth goal in seven internationals calmed the nerves inside Wembley only momentarily as Hart brilliantly turned away Damjanovic's goal-bound header.
A booking for Kyle Walker, ruling him out of the Poland qualifier, cast a shadow over the start of the second half but England soon had a second goal to smile about as Boskovic put Welbeck's cross into his own net from fully 15 yards.
Relief was almost replaced by trepidation once again as Jovetic's effort clattered against the crossbar as Montenegro tried to vein to give their World Cup hopes a lifeline.
Gerrard missed a virtual open goal when set up by Rooney and Poleksic denied Welbeck from close range and those misses threatened to haunt England when the visitors halved the defect as Demjanovic poked home from Jovetic's shot.
But nerves were firmly calmed 12 minutes from time as Townsend slammed home from outside the box and then Sturridge won and then converted a late penalty to guarantee England at least a play-off place and put them on the brink of qualification.
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