The Midlanders made a great start, leading 5-2 after the first period with goals by Jordan Elliot (2), Peter Farkas, Matt Kelly and Nick Beard. Cheltenham slowly battled their way back into contention, reducing the lead to 7-4 after two periods and 7-6 after three.
By this stage, Cheltenham were dominating much of the play - but missing a number of clear cut chances. They paid the penalty when Ryan Padbury made it 8-6 to Solihull, followed by another Cheltenham goal by Adam Legge.
Again, Cheltenham fluffed their lines when chances were on offer. And Solihull were able to celebrate a notable victory when Morgan Hirsch netted on the final whistle.
Earlier, Manchester were too strong for Sheffield, winning 10-3. Tom Curwen, the League's top marksman on a number of occasions, began with a five-goal haul in Lancaster's 14-12 defeat of Invicta.
In men's Division 2, Sutton and Cheam rallied too late from a five-goal deficit and went down 10-9 to derby rivals West London Penguin. For Penguin, there were two goals apiece for Bence Kadar, Lazzaro Vallarino and Dragos Ungureanu.
Sam Smith and John Pratt both claimed hat-tricks for Team Northumbria, who beat Welsh Wanderers 9-8. Northumbria held a 9-5 advantage at the start of the final period and took the points despite leaking late goals by Dan Laxton (2) and Charlie Hinder. Outstanding left-hander Kameron Powell was on target five times in Exeter's 13-10 victory over Croydon.
Chris Icely (5) and Chris Brown (4) were the main scorers in Polytechnic's 10-6 defeat of Worthing.