The victory for the Salalah-based giant gave it the sole lead at the top of the table after seven rounds with 17 points.
Dhofar took the lead through Lawson's goal in the 20th minute and was reduced to ten men when its goalkeeper Riyadh Subeit was given the marching orders in the 62th minute.
Despite being a player short, Dhofar scored its second goal in the 73rd minute when Lawson found the net again.
Al Shabab, which went into the seventh round with the same number of points as Dhofar and trailed on goal-difference, fell behind after it was held to a 1-1 draw by former champion Al Nahda on Sunday.
The draw meant Shabab remained second with 15 points, while Sohar and Al Arouba are third and fourth.
Both teams have 14 points each, though. Arouba climbed to the fourth place after it won 3-1 against Muscat at the Sur Sports Complex on Sunday, while Sohar wasted an opportunity to climb up after it was held to a 1-1 draw by defending champion Fanja at the Seeb Sports Stadium on Sunday.
Fanja, which has had a disappointing start to its defence, lost three of its seven games, managing only two wins.
The latest draw kept it in the mid-table as it is in eighth place with eight points - the same as Saham, with the latter ahead on goal-difference. Saham went down to Al Nasr 0-2 on Saturday in its home game in Sohar.
Al Nasr is in fifth place with 12 points, followed by Oman with 11 points. Oman was held to a goalless draw by Jalan in Sur, and in a Batinah derby, former champion Al Suwaiq was held to a 1-1 draw by OPL debutant Rustaq, leaving both teams just above the relegation zone.
Standings (after seven rounds): 1 Dhofar (17 pts), 2 Al Shabab (15), 3 Sohar (14), 4 Al Arouba (14), 5 Al Nasr (12), 6 Oman (11), 7 Saham (8), 8 Fanja (8), 9 Al Nahda (7). 10 Al Khabourah (7), 11 Rustaq (6), 12 Al Suwaiq (6), 13 Jalan (3), 14 Muscat (2).