The defender rifled home a powerful shot after 31 minutes before late goals from Shayne Lavery and Keshi Anderson wrapped up the tie.
Blackpool should have won even more comfortably after seeing a host of chances go begging in the all-Sky Bet Championship clash.
James Husband, Josh Bowler and Connolly all went close inside the opening half hour but Connolly finally put the home side ahead when he smashed home after a Bowler free-kick had hit the wall.
Martin Payero was off target with a free-kick as Middlesbrough finally registered their first attempt three minutes later.
Bowler had a shot saved by Joe Lumley after a good run on the stroke of half-time, while Lavery was denied and Bowler curled an effort wide as Blackpool continued to dominate proceedings.
Isaiah Jones fired over for the visitors after a good run in the 71st minute.
But Lavery wrapped up the tie with his second goal in two games after beating Lumley at his near post in the 77th minute and substitute Anderson made it 3-0 a minute later after his flick-on looped in at the back post.
Middlesbrough manager Neil Warnock said: “We got well and truly beaten.
“At 1-0 I thought we had some good chances to get back in it, but we never really capitalised on anything. We never looked like scoring.
“I am not very happy tonight. It is disappointing because that is our back-up squad against their back-up squad.
“It has answered a few questions, which is what I wanted. We have not got a squad big enough to come here and lose two players.
“Tonight shows what we have got in back-up and it is disappointing really. There were one or two players that were poor.”
Blackpool: Grimshaw, Connolly, Ekpiteta, Casey, Husband, Bowler, James, Dougall (Ward 46), Hamilton (Anderson 63), Carey, Lavery (Yates 81). Unused substitutes: Moore, Keogh, John-Jules, Garbutt.
Middlesbrough: Lumley, Peltier, Wood-Gordon, Bamba, Robinson, Malley, Morsy (Sivi 63), Jones, Spence, Coburn (Ikpeazu 63), Payero (Hackney 71). Unused substitutes: Anfernee Dijksteel, Stojanovic, Malik Dijksteel, Sykes.
Referee: R Madley (West Yorkshire).