He was taken off on a stretcher in the 76th minute after falling awkwardly under a challenge.
The Czech goalkeeper was taken to hospital with a suspected broken arm and manager Dennis Wise fears he may be without his first-choice shot-stopper for at least two months if his suspicions are confirmed.
Swindon put defender Ady Williams in goal for the last 14 minutes of normal time and for time added on for stoppages but Grimsby were unable to test him.
The Robins were coasting after Michael Pook scored his first league goal for the club in the 31st minute after rounding off a slick move.
And from that moment Grimsby did not mount a serious threat to the outcome.
Defensive confusion allowed Swindon to score again in the 44th minute when Andy Monkhouse chased through to score and Swindon were able to control the second half without being at their best.
Striker Lee Peacock added a third near the end but by then Grimsby were a well-beaten side.
Wise chose to leave reserve goalkeeper Phil Smith off the bench so that he could play an extra attacker but he need not have worried about Williams when the defender donned the goalkeeping jersey.
Grimsby had their moments but struggled to make an impact, to the annoyance of manager Graham Rodger who paced the technical area demanding greater effort from his players.
There might have been more for Swindon with Peacock going close on two occasions and new signing Alex Rhodes almost marking his debut with a goal.
Rhodes, on a month's loan from Brentford, had the ball in the Grimsby net but the effort was disallowed for a marginal offside.
Rhodes caused Grimsby plenty of trouble as Swindon ended a run of four successive draws and five matches without a win.
Williams had been a solid figure at the heart of the Swindon resistance before he was called on to go in goal and although Grimsby did their best to unsettle him with crosses into the box, he dealt with everything competently.