Trailing early on to Wilfried Bony's first goal since January, the Tigers spurned a lifeline from the spot six minutes before the break when Bowen's effort was superbly turned away by Swansea keeper Erwin Mulder.
But the winger dusted himself down and redeemed himself with a 70th-minute leveller from close range - to trigger a golden run of three goals in the space of ten minutes to turn the game on its head.
Tommy Elphick's header six minutes later - his first goal for the club - put Hull in front before Bowen went solo before netting in eye-catching fashion, to take his season's tally to eight.
A late goal from substitute Bersant Celina set up a nervy finale, but the hosts held on for a precious win to continue their resurgence and move them towards the safe confines of mid-table.
The hosts barely laid a glove on a slick Swansea outfit high on clever movement and comfortable in possession in the first half an hour - only for the penny to finally drop with the hosts much improved in the final 15 minutes.
Passing up a penalty gift was careless, even if Mulder produced a fine save to turn away Bowen's strike after he raced out and caught Fraizer Campbell, but the hosts' response was stirring.
It was far removed from the bulk of the first half, which was one-way traffic.
Swansea, backed by a small enclave of supporters who were faced with negotiating an epic 590-mile round trip, were dominant from the off and controlled the opening half hour.
A crisp third-minute low drive from Bony, which went in off the post, gave the Swans the perfect platform and they manufactured several other chances - which could have put the game out of sight.
Neat play resulted in Matt Grimes testing the reactions of David Marshall before good work by Barrie McKay ended with Bony firing acrobatically over from close range.
Hull were soon unhinged again, with Connor Roberts picking out Nathan Dyer, who fired over when well placed before Todd Kane got in the way of a goalbound effort from Bony.
The Tigers finally started to turn the tide and announced themselves on proceedings when Kamil Grosicki was off target following a fine cross by Bowen - in a much-needed moment of assertion from the hosts.
The winger was centre stage soon after, assuming penalty duties after Campbell was felled, only to see his effort splendidly saved.
Soon after, Mulder again excelled himself to block a close-range effort from Grosicki following a brilliant run as Hull ended the half strongly.
The interval came at a good juncture for Swansea, who rediscovered their first-half vim on the resumption and displayed danger going forward once again.
A sweet move involving Fer and Roberts ended with the unmarked Wayne Routledge dragging a low shot wide - as the Swans' wastefulness in front of goal was again showcased.
A flashpoint soon ended with Campbell and Mike van der Hoorn both booked with Swansea continuing to hold the hegemony in terms of possession, with Hull losing their way again.
In a bid to wrestle back the initiative, Evandro was brought on midway through the second half for his first appearance since September 18 following injury and not too long after, the Tigers were level.
Bowen was afforded his relieving moment from close in, netting after good work by Grosicki and Campbell.
The goal provided a badly-needed infusion of confidence to Hull and soon after, those in amber and black were on their feet again.
Grosicki's hanging corner was attacked by Elphick, with his fine looping header flying over Mulder.
Hull went for the jugular with Swansea increasingly besieged and a classy third duly arrived with Bowen surging clear on the right before producing a cool steered finish past Mulder - to complete a dream spell for the hosts.
A late goal from Celina set off a few flutters before fellow replacement Chris Martin almost sealed it, with the final action seeing Oli McBurnie head just wide.
Hull City: Marshall; Kane, Elphick, Mazuch (de Wijs 45), Kingsley; Stewart, Batty (Evandro 66); Bowen, Irvine, Grosicki; Campbell (Martin 85). Substitutes unused: Long, Dicko, Fleming, McKenzie.
Mulder; Roberts, Rodon, Van Der Hoorn, John (James 77); Fer, Grimes; Routledge, McKay, Dyer (Celina 77); Bony (McBurnie 64). Substitutes unused: Nordfeldt, Carter-Vickers, Naughton, Fulton.
Referee: O Langford (West Midlands).