The Spanish international had netted just three times for the Premier League strugglers, failing to score in 13 of his previous 15 appearances, since his summer signing.
But the 31-year-old forward was on target after 18 minutes at the Riverside off an Adam Clayton cross and 11 minutes later converted Boro's first penalty of the season after Jordi Amat tripped up Adam Forshaw in the box.
Dutch midfielder Marten de Roon added a third on 58 minutes off a pin-point Gaston Ramirez cross as Boro rallied to just their fourth win since promotion from the Championship.
Middlesbrough moved above champions Leicester to 14th place as Swansea dropped to second-bottom.
The defeat piles more pressure on the Welsh outfit who headed north without Federico Fernandez and Ki Sung-yueng, who were both recovering from toe injuries.
Bob Bradley's side had gotten off to a bright enough start with Modou Barrow having a cross cut out by Ben Gibson and a Gylfi Sigurdsson shot kept out by Boro keeper Victor Valdes.
Negredo had his first chance in the 10th minute but his effort was cleared by Swans goalkeeper Lukasz Fabianski.
Danish midfielder Viktor Fischer hobbled off for the hosts with a knee problem early but Negredo grabbed the opener when he finished off a Clayton cross with Ramirez almost grabbing a second minutes later from a free-kick after being fouled by Angel Rangel.
Boro doubled their account before the half-hour mark when Amat tripped Forshaw in the box. Negredo made no mistake to ensure the home side ended the first period well in control.
Despite the odds stacked against them Swansea continued to press forward with Sigurdsson firing over the bar and Leroy Fer effort blocked.
But Swansea's hopes of a late consolation goal were dashed when Valdes saved from Fernando Llorente late, as the visitors slumped to second from bottom ahead of Hull on goal difference.