In atrocious conditions at Bloomfield Road, with a swirling wind and driving rain, the Town scored once in the first half and twice after the break to bring the recent run of indifferent form to an end and move back up to 12th in the Championship table.
Manager Jones had shuffled the pack and made five changes from the side that lost 2-1 at home to Cardiff in the previous outing.
One change was enforced, with James Shea coming back in for Simon Sluga, who did not travel north with the squad due to illness. Elsewhere Carlos Mendes Gomes made only his second league start supporting Adebayo in an attacking role.
Allan Campbell and Henri Lansbury were brought back into midfield and Reece Burke came in for Tom Lockyer on the right side of the back three.
Shea was called into action in the sixth minute, saving low at his near post from Owen Dale, before the whole stadium joined in with the rest of English football in a minute's applause for tragic Birmingham City supporter Arthur Labinjo-Hughes, the six-year-old whose murder has shocked the nation this week.
The Hatters soon went on the offensive, Lansbury firing a shot over, Clark seeing one deflected wide and Adebayo heading straight at keeper Chris Maxwell from a whipped in James Bree cross, all by the 11th minute.
Mendes Gomes was getting himself involved, and in the 16th minute he created a chance for Adebayo, who sprang the Blackpool offside trap to latch on to the former Morecambe man's through pass, but sent his low shot into the side-netting from 15 yards.
The post came to Town's rescue after Shea had saved Reece James' left-footed drive in the 22nd minute, Demetri Mitchell's follow-up deflecting off Bradley's heel and off the base of the upright to safety.
Shea was at his best again just before the half-hour, flying across his line to tip a 25-yarder from Ryan Wintle over the bar, with Adebayo producing three big headers clear from the resulting corner to avert the danger.
The former Arsenal and AFC Wimbledon keeper showed his agility again in the 31st minute, leaping to his left to push a header from Dale around his left post, after being impeded on his goal-line by a Blackpool attacker as he looked to come for the initial ball into the box.
The Town had weathered a storm, both literally and in a footballing sense, and then took the lead with half-time approaching.
Clark was fouled 35 yards from goal and Bree arced a free-kick towards the Blackpool goal that kicked up off the greasy surface, forcing Maxwell to push it behind for a corner.
Naismith delivered and captain Bradley rose highest at the far post to head downwards, with the ball bouncing up and going in off the underside of the cross bar.
Jones made a change at the break, bringing Pelly-Ruddock Mpanzu on for Lansbury in midfield, but it was the hosts who had the first attempt of the second half, Jerry Yates finding space on the edge of the box but seeing his shot deflected over the bar.
By the 53rd minute the lead was doubled. Clark, who had impressed as the Hatters' stand-out player to that point, twice tried to shoot, only to see them blocked almost before they left his boot.
He retrieved the ball wide on the left, then floated a lovely cross to the far post where Adebayo was loitering to nod home a simple ninth goal of the season.
Blackpool tried to mount a response, but Shea comfortably gathered a Kenny Dougall shot that looped up from a deflection off Burke, then Wintle's attempted curler went well wide.
The superb Shea had to tip another over the bar on 71 minutes, James' shot from the edge of the box deflected off Bree and looking to be heading for the top corner, then Naismith threw himself in front of a fierce half-volley from sub Shayne Lavery as the Town backline stood strong, with the Scot, skipper Bradley, Burke, Bree and Bell all making vital interceptions.
The icing on the cake came a minute into time added on, when sub Admiral Muskwe cut the ball back from the byline for Clark to steer home a deserved second goal in as many games, and send the 1,391 travelling Hatters off home - or into Blackpool - in celebratory mood.
Goals:
Town - Bradley 42, Adebayo 53, Clark 90+1
Att: 11,366 (1,391 Hatters)