Magpies winger Jonah Lowe opened the scoring inside five minutes in Napier, but the hosts went into the break down 13-8. Weber's double in quick time early in the second half put the hosts back in the lead, but tries to Judd and Mike Delany put the visitors back ahead.
When Delany slotted a penalty to put his side ahead by six points with 10 minutes remaining, Bay of Plenty looked set to wrap the match up. Instead, they were left dumbfounded as to how they could allow Apikotoa to barge his way over the line from close range near the sticks; an easy conversion for Tiaan Falcon clinching the win.
Hawke's Bay remain at the top of the Championship ladder with the win – moving seven points clear of Otago in second.
Southland remain anchored to the bottom of the ladder without a win, after being decimated 43-26 by Counties Manukau in the earlier match in Invercargill.
The hosts started the match in the best way possible, building a 14-0 lead within 22 minutes. It wasn't enough of a headstart however, as Counties Manukau kicked into gear around the 25 minute mark, scoring six tries before Southland were able to get back across the line.
Hawke's Bay 29 (Brad Weber 2, Joe Apikotoa, Jonah Lowe tries; Tiaan Falcon 3 cons, pen)
Bay of Plenty 28 (Richard Judd 2, Mike Delany tries; Delany 2 cons; 3 pens).
HT: 8-13
Counties Manukau 43 (Fotu Lokotui, Luteru Laulala, Te Aihe Toma, Nigel Ah Wong, Joe Royal, Cardiff Vaega tries; Latiume Fosita 4 cons, Laulaua con, pen)
Southland 26 (Bill Fukofuka, Morgan Mitchell, Lewis Ormond, Nico Costa tries; James Wilson 3 cons)
HT: 19-14