Kabelo Dlamini put Celtic ahead just before half-time and Zimbabwean Knox Mutizwa levelled midway through the second half with both scoring their fourth league goals of the season.
Former South Africa captain and centre-back Komphela quit cash-strapped Celtic in December and was immediately snapped up by Arrows, who had just fired Clinton Larsen for poor results.
In a resignation letter leaked to the media, Komphela listed numerous complaints, including that Celtic regularly did not pay his monthly TV sport channel subscription.
"This was a chess match," the coach said about the Bloemfontein draw that kicked off in 35 degree celsius (95 fahrenheit) heat. "I had to constantly change my tactical approach."
Former teacher Komphela is popular for his post-match quotes, including: "Statistics are like a bikini, they do not reveal everything."
Despite being held at home, Celtic rose two places to fifth after 17 rounds as they seek a first Premiership title while Arrows remained 10th.
Celtic are for sale with owner Max Tshabalala admitting he can no longer fund the approximate four million rand ($290,000, 250,000 euros) needed each month to run the club.