Boro's first real attack of the game brought a free-kick for the visitors on the edge of the area. Adam Johnson stepped up, bent it round the wall and into the bottom corner past former Chelsea goalkeeper Lenny Pidgeley to give Boro the lead on four minutes.
A surge, spanning 20 yards, from substitute Williams saw him beat three Carlisle players before lashing the ball from 15 yards past sub keeper Adam Collin in the Cumbrians' goal to double Boro's advantage with 10 minutes remaining.
Middlesbrough: Jones (Coyne 60); McMahon, Huth, Wheater (Hoyte 60), Taylor (Williams 72), O'Neil, Arca (Capt) (Hines 60), Bates (Walker 25), A Johnson (Martin 83), Yeates (Porritt 72), Franks. Unused subs: J Johnson, Smallwood.
Carlisle United: Pidgeley (Collin 74); Raven (Kane 46), Keogh (Livesey 46), Murphy, Harte (Horwood 46), Taylor (Rothery 46), Thirlwell (Burns 72), Taiwo, Robson (Tinnion 74), Anyinsah (Madine 56), Dobie (Bridge-Wilkinson 56).
Referee: Clive Oliver, Northumberland.