The Saints now face Biarritz at Franklins Gardens next weekend in the showdown to decide who will progress as group winners.
Reivers had made a stirring start, grabbing the lead in the seventh minute thanks to a try by hooker Ross Ford, scuttling down the narrow side after a lineout clutch from Opeta Palepoi.
The score acted as a wake-up call to the Saints, who hit back almost immediately as slick work by John Howard and Carlos Spencer created the time and space for Paul Tupai to reach the line.
Former All Blacks player Spencer converted, then banged over a penalty to give his side the momentum.
However, Borders snatched back the edge when winger Simon Danielli galloped almost 80 metres for an interception touchdown, converted by skipper Calum MacRae.
The referee sent former Scotland team-mates Stuart Grimes and Tom Smith to the sin-bin for fighting.
MacRae was then wide with a penalty attempt, and Saints made the most of the let-off, with Spencer’s dazzling sidestep leading to their second try by full-back Paul Diggin. Spencer added the extras.
Any prospect of a Borders fightback was wiped out by England’s Ben Cohen, who bagged two trademark touchdowns in seven minutes after the restart, the first of which was converted by Spencer.
The brave Reivers refused to throw in the towel and they rallied to set up a score for MacRae, who also supplied the accurate kick.
Northampton coach Paul Grayson denied afterwards he had demanded the controversial last-gasp switch from Netherdale to Murrayfield.
And Grayson said: “I was more than prepared to play — wherever the game was to be. It was entirely up to the referee (George Clancy).”
On his side’s display, Grayson added: “I was particularly delighted with the performance of Ben Cohen at centre. In addition to his two tries, he put in a power of work.”
Tries: Ford, Danielli, MacRae. Cons: MacRae 2.
Tries: Tupai, Diggin, Cohen 2. Cons: Spencer 3. Pens: Spencer.
MacRae, Danielli, Law, Rennie, Daniel, Townsend, Chrystie, Kerr, Ford, Douglas, Grimes, Palepoi, Newlands, Miller, Brown.
Jones for Rennie (25), Robertson for Daniel (11), Kalman for Kerr (41), Vernon for Palepoi (73), Dalziel for Newlands (52).
Diggin, Lamont, Cohen, Kydd, J. Howard, Spencer, Robinson, Smith, Thompson, Barnard, Short, Gerard, Tupai, Fox, D. Browne.
Laird for Diggin (75), Lord for Lamont (67), Hartley for Thompson (65), Lewitt for Fox (62), Easter for D. Browne (77).
George Clancy (RU).