Rovers saw off Kennoway Star Hearts 2-0 at Whitestone Park and the Streamers edged out Burntisland Shipyard 1-0 at Home Park.
Grant Wilson scored both of Peebles’ goals, one on seven minutes from the penalty spot and the other just ahead of half-time.
Coldstream’s goal was scored by Miguel Freire with 82 minutes on the clock.
Saturday was less successful for the region’s EoSFL first and third division sides, with Vale of Leithen losing 2-0 away to Ormiston Primrose in the King Cup’s second round and Hawick Royal Albert and Linton Hotspur being beaten at home on league duty.
Those third division defeats were by 2-1 for Albert hosting Stoneyburn and 2-0 for Hotspur versus Bathgate Thistle.
Pol Adrian and Jacob Riley scored for Ormiston on 32 minutes and 81 respectively against a Vale side reduced to nine men by red cards for Harris Neillings not long after the half-hour mark and Gary Mitchell five minutes ahead of the final whistle.
Davie Scholtes scored for Albert on 73 minutes, with Liam Woods and Joshua Sharpe netting for their visitors on 53, from the penalty spot, and 90 respectively.
Bathgate’s goals in West Linton were scored by Ally McInnes and Glenn Thomson.
All five of the region’s EoSFL sides are on league duty this coming Saturday – Vale, Peebles, Coldstream and Hotspur to Lochore Welfare, Thornton Hibs, Newburgh and West Calder United respectively and Albert away to Pumpherston, all 2.30pm kick-offs.
Next Tuesday brings three further league fixtures – Vale at Preston Athletic, Coldstream hosting Tweedmouth Rangers and a Borders derby for Albert at home to Hotspur.