Mowatt was on hand to fire home just three minutes into a pulsating content, with Leeds somehow managing to hold onto that advantage for the rest of the encounter and claim their first double over Boro since 1989-90.
Some truly heroic defending continually denied Boro, unbeaten in their previous 12 home matches and seeking a sixth successive Riverside success.
When the back four was breached, Boro found a competent last line of defence in Marco Silvestri, who thwarted the hosts’ at several key junctures.
Mowatt’s sixth goal of the campaign helped Leeds become the first side to triumph at the Riverside since Reading won by an identical single-goal margin on August 30.
The win also extended Leeds’ excellent sequence at the Riverside, where they have now been beaten just once in 13 matches at the north-east venue.
The win was Leeds’ fifth in six outings, with Neil Redfearn’s side having travelled the equivalent of a million miles in a footballing sense since their previous visit to a promotion-chaser in Derby County on a bleak end to 2014 on December 30.
Leeds’ resolution, so lacking at Derby, was immense, although on another day, Boro could have been easily celebrating victory with nothing quite falling their way.
But credit was due to the battling qualities of Leeds, who look a side transformed this calendar year, which has since them claim noteworthy doubles over the top two of Bournemouth and now Boro.
The work ethic throughout the Leeds line-up was magnificent and while luck went their way on a few occasions, they certainly earned it.
The tone was set in a first-half in which Boro laid siege on goal at times, but United’s lead remained intact, thanks to a combination of some stoic last-ditch defending, impressive saves from Silvestri and some profligate finishing from the hosts.
While the majority of the play was in the Leeds half, with Boro posting all manner of efforts of goal, Leeds could have taken in a two-goal lead, with Sol Bamba spurning a great chance to put them 2-0 ahead moments before the break following Luke Murphy’s corner.
United, unchanged from the side who beat Millwall seven days earlier, were afforded the perfect start with a third-minute opener to rock the majority of the Riverside Stadium.
Tidy build-up on the right saw Cook burst forward and his intelligent cut-back was seized upon in a flash by Alex Mowatt, whose low shot whistled past Tomas Mejias, with the home keeper barely having time to react.
It ensured Boro experienced the rare sensation of being behind on home soil for the first time since way back on August 30, but their reaction was swift.
A smart free-kick from skipper Grant Leadbitter saw found Jelle Vossen, playing just behind lone forward Kike, but his fresh-air shot when well placed summed up a frustrating half for the Belgian and table-topping Boro.
Bamba had to be alert to deny Kike before the Spaniard soon saw his effort blocked by Silvestri after being played in by Albert Adomah.
Leeds possessed poise on the break in Cook, with Rudy Austin’s energetic thrusts also hinting at danger, but it was a half when the visitors sought largely to protect their precious gains.
Silvestri was forced to show his mettle again on 19 minutes when he turned away a well-struck near-post volley from Vossen after good play on the right by Emilio Nsue.
Boro soon went desperately close to a leveller, when Adam Reach’s cross-shot found Vossen, who couldn’t get a telling connection to divert the ball into the net, with Giuseppe Bellusci clearing in the nick of time.
Some of Boro’s build-up was of surgical precision, but Leeds survived the onslaught just about.
Leadbitter fired wide after more neat build-up before Kike’s shot on the turn was tipped over adeptly by Silvestri, who earned his corn during the first half.
At the other end, a good move involving Cook and Sam Byram found Steve Morison free down the left, but his cut-back couldn’t be dispatched by a visiting player before play switched back to the Leeds half.
A strike from ex-United midfielder Adam Clayton was then turned away before the half ended with a great chance to Leeds when Bamba headed wide in splendid isolation following Murphy’s corner.
The pattern of most of the first half continued on the restart, with Kike denied by Silvestri before a free-kick on the right almost resulted in a second for United, when Murphy’s delivery was met by Scott Wootton, whose header cannoned off the bar.
Kike then saw his low shot turned away with the impression that it might not be Boro’s day starting to generate.
A cross-shot from Adomah was soon tipped away by Silvestri, but while Leeds’ attacks were infrequent, they still possessed danger with Boro’s backline not looking wholly convincing.
A clearance from Ben Gibson denied Mowatt in the nick of time, with Leeds mindful that a second goal would effectively kill the game, with their threat on the break, if not exactly potent, at least prevalent.
Boro threw on Lee Tomlin on 57 minutes, with top-scorer Patrick Bamford also entering the fray seven minutes later, with all of the hosts’ attacking armoury now in the mix.
A brilliant challenge from Bamba, outstanding at the back, denied Kike in the nick of time as Boro launched another assault on the Leeds rearguard.
Vossen’s shot was then turned away by Silvestri before a saving challenge by Charlie Taylor thwarted Leadbitter.
As the clock ticked, so a sense of dispondency started to set in among home supporters in the 25,531 crowd with Leeds going close to sealing it from an unlikely source when Wootton arriving at the far post, couldn’t convert a low cross from Byram.
The seal on a difficult afternoon for Boro arrived late on when Vossen came off second best in a challenge in midfield with Cook, with the Boro forward receiving lengthy attention before being stretchered off.
It left Boro to play out the game with ten men, having used all their substitutes.
After a long delay, Leeds were left to negotiate nine minutes of added-on time, but it was they who come closest to another goal when Austin’s shot was tipped over.
Middlesbrough: Mejias; Nsue (Bamford 66), Kalas, Gibson, Friend; Leadbitter (Forshaw 79), Clayton, Adomah, Reach (Tomlin 57); Vossen, Kike. Substitutes not used: Ripley, Husband, Whitehead, Omeruo.
Leeds: Silvestri, Wootton, Bamba, Bellusci (Cooper 90), C Taylor, Byram, Austin, Murphy, Cook, Mowatt, Morison. Substitutes not used: S Taylor, Berardi, Sloth, Sharp, Cani, Antenucci.
Referee: Andre Marriner (West Midlands).
Attendance: 25,531 (2,690 Leeds fans).