The 32-year-old Kapur carded a fine six-under 66 to move to seven-under 137 and shared the lead with Pavit Tangkamol Prasert of Thailand.
Right behind Kapur and Pavit were Arjun Atwal (73-65), the Asia No. 1 in 2003, and Gaganjeet Bhullar (69-69) in tied third with Simon Yates ((71-67) at six-under.
Atwal (73-65), who has seven wins on Asian Tour, turned in one of his best rounds in recent months with a flawless seven-under 65 on Friday. Countryman Gaganjeet Bhullar (69-69) moved to six-under.
"I like Dubai, it is my second home," said Kapur with a big smile. "I'm a resident of Dubai, have lots of friends and family here. I love this place. The last time I won here it was the season ending event (on European Challenge Tour), I won, so fingers crossed I can continue the mojo."
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Jeev Milkha Singh (71) was even par after 36 holes, while Daniel Chopra fired two rounds of 72 each to be tied with Jeev at 46th.
Jyoti Randhawa (74 and 73), Chiragh Kumar (73-75) who closed with a double bogey, will miss the cut, as will SSP Chowrasia (75-73) and Rashid Khan (75-74).
Since his historic triumph on the PGA Tour in 2010, Atwal's form has dipped due to injuries and he has notched only one top-10 over the past two years. But with the Dubai Open being his fifth straight tournament on the Asian Tour, where he has not missed a cut, he hopes to turn the corner with his game.