In a good opening game both sides played attractive football and the first real chance fell to Ricketts in the 16th minute, but the former England striker parried wide from a free-kick by Marco Reich.
Ricketts atoned just two minutes later notching his tenth goal of the season from 12 yards after Nicholls and Mark Bradley had both had efforts blocked.
After that it was largely the Robins who dominated, with striker Billy Paynter a constant threat particularly in the air.
In the 28th minute he had a header well saved by Gilmartin following a good corner by Jon-Paul McGovern.
Swindon always looking likely to score, went close again when Simon Cox brought the ball down brilliantly from a pass by McGovern only to fire over.
Ricketts, who had a first half appeal for a penalty turned down, failed to appear for the second half, replaced by Frenchman Sofiane Zaaboub.
But it was the visitors who had most of the play, although Walsall's Troy Deeney had the ball in the net on 65 minutes but was dubiously ruled offside.
In the 71st minute Paynter had another free header from a cross by McGovern but Gilmartin saved splendidly with one hand.
Seconds later Cox thought he had equalised for Swindon when netting with a low header, but the flag was up for offside.
The Robins looked almost certain to equalise, but Gilmartin saved well from Lee Peacock in the 82nd minute and pulled off an even better save from Cox just a minute later.
Walsall threatened only rarely, but they increased the lead in the final minute when Nicholls scored with a close-range header from a cross by substitute Richard Taundry.
With the game into added time Swindon broke again and Paynter gave them hope by scoring with a speculative shot from 30 yards.
But it was Walsall who finished slightly the stronger and Deeney had a shot saved by Phil Smith.