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Borough travelled in confident mood following their recent upturn in form, which included victories over Coggeshall Town and AS London, and made a positive start as they looked to build on that momentum.
Their attacking approach was rewarded after 25 minutes when a well-worked move created an opening.
The initial effort struck the post, but Wilson Chingoka reacted quickest to the rebound and converted from close range to put the visitors ahead.
Enfield continued to push forward in search of a second but had to settle for a slender 1-0 advantage at the interval.
Basildon responded after the restart and drew level from a set-piece.
Borough failed to deal with the second ball after conceding a free-kick in a dangerous area and a home attacker reacted quickest to score from close range.
Olu's side refused to let the setback derail them and increased the tempo as they sought to regain control.
A slick passing move opened up the Basildon defence and released Remel Green into the penalty area.
A well-timed challenge prevented Green from getting his shot away, but the loose ball fell invitingly to Monet Ollivierre.
Ollivierre showed excellent composure, guiding a left-footed finish into the bottom corner to restore Borough’s advantage at 2-1.
Having regained the lead, Enfield looked well placed to secure a third successive league victory, but Basildon responded once again with another equaliser.
Borough pushed for a winner in the closing stages but were unable to make the decisive breakthrough, leaving the visitors to settle for a share of the spoils.
While an away point represented another positive result following a difficult start to the campaign, there was understandable frustration after Enfield had twice put themselves in front.
Nevertheless, the performance provided further encouragement, particularly in the quality of the attacking moves which produced their two goals.
The draw extends Borough’s unbeaten run to five matches and continues the progress shown during recent weeks.